Algebra Quiz: Interpreting Sketching Key Features Of Functions

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Interpreting Sketching Key Features Of FunctionsQuestion 1 of 20

A roller coaster's height (in meters) above the ground is modeled by $H(x)=-(x-2)^2+9$ , where $x$ is the horizontal distance (in meters) from the start of a section of track. What is the maximum of $H(x)$ ?

Maximum at

(2,-9)

Maximum at

(2,9)

Maximum at

(-2,9)

Maximum at

(9,2)

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Algebra Quiz

Algebra Quiz: Interpreting Sketching Key Features Of Functions

Practice Interpreting Sketching Key Features Of Functions in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting Sketching Key Features Of Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Maximum at $(2,-9)$
Maximum at $(2,9)$ (correct answer)
Maximum at $(-2,9)$
Maximum at $(9,2)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. A maximum is the highest point on a graph (or on some portion of it), giving the largest y-value, while a minimum is the lowest point, giving the smallest y-value. For a parabola that opens up, the vertex is the minimum; if it opens down, the vertex is the maximum. In real-world problems, these tell you the best or worst outcome! For this downward-opening parabola H(x) = -(x-2)^2 + 9, the vertex at (2,9) is the maximum, meaning the highest point on the roller coaster track is 9 meters at x=2 meters horizontally. Choice A correctly identifies the maximum as (2,9) because the vertex form shows the peak at x=2, H(2)=9. Choice D identifies the maximum but at the wrong location: (-2,9) might come from misreading the vertex, but it's at x=2—make sure you're finding the right extreme! Quick reference for graph features: (1) Intercepts—where graph crosses axes, write as points (x, y), (2) Increasing/decreasing—read left to right, going up = increasing, going down = decreasing, state as x-intervals, (3) Positive/negative—above x-axis = positive, below = negative, state as x-intervals, (4) Maximum/minimum—highest/lowest points, give as points (x, y) or just y-value if asked, (5) End behavior—what happens at far left and far right of graph.

Question 2

A ball is thrown upward from a platform. Its height above the ground after $t$ seconds is modeled by $h(t)=-5t^2+20t+15$ , where $h$ is in meters. What is the y-intercept of $h(t)$ , and what does it represent in context?

$(0,15)$ ; the initial height of the ball at $t=0$ (correct answer)
$(15,0)$ ; the time when the ball hits the ground
$(0,-15)$ ; the height below ground at $t=0$
$15$ ; the time when the ball is thrown

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' In this context where the function models the height of a ball thrown upward over time, the y-intercept at (0, 15) means the initial height of the ball is 15 meters when t=0, before any time has passed. Choice B correctly identifies the y-intercept as (0,15) and interprets it as the initial height of the ball at t=0 because plugging t=0 into h(t) gives h(0)=15, representing the starting point. Choice A confuses the y-intercept with an x-intercept: (15,0) would be where height is zero, like when the ball hits the ground, but that's not the y-intercept—always check by setting the input to zero for y-intercept! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!

Question 3

A population of bacteria (in thousands) after $t$ hours is modeled by $B(t)=3\cdot 2^t$ . What does the $y$ -intercept represent in this context?

The time when the population reaches 0
The initial population at $t=0$ (correct answer)
The population after 1 hour
The time when the population doubles

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' For B(t) = 3·2^t, the y-intercept occurs when t = 0. Substituting: B(0) = 3·2⁰ = 3·1 = 3. Since B(t) represents population in thousands and t represents time in hours, the y-intercept of 3 means there were 3,000 bacteria at time t = 0—the initial population when observations began. Choice B correctly interprets the y-intercept as the initial population at t = 0 because the y-intercept always represents the function's value when the input variable equals zero, which in time-based contexts means the starting condition. Choice C might seem tempting since it mentions 'after 1 hour,' but that would be B(1) = 3·2¹ = 6, not the y-intercept—the y-intercept specifically requires t = 0, not t = 1! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). In exponential growth/decay problems, the y-intercept is almost always the initial amount before any growth or decay has occurred. Always substitute x = 0 (or t = 0) to find the y-intercept!

Question 4

Describe the end behavior of the function $f(x)=-3x^3+2x$ .

As $x\to\infty$ , $f(x)\to\infty$ and as $x\to-\infty$ , $f(x)\to-\infty$
As $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to\infty$ (correct answer)
As $x\to\infty$ , $f(x)\to 0$ and as $x\to-\infty$ , $f(x)\to 0$
As $x\to\infty$ , $f(x)\to\infty$ and as $x\to-\infty$ , $f(x)\to\infty$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For f(x) = -3x³ + 2x, the end behavior is determined by the leading term -3x³. Since this is a negative odd-degree term, as x → ∞ (x gets very large positive), -3x³ → -∞ (very large negative), and as x → -∞ (x gets very large negative), -3x³ → ∞ (very large positive). The +2x term doesn't affect end behavior because x³ grows much faster than x. Choice B correctly states 'As x→∞, f(x)→-∞ and as x→-∞, f(x)→∞' because the negative cubic function goes down on the right and up on the left. Choice A reverses this behavior—that would be true for a positive cubic like +3x³, not a negative one! End behavior shortcut for polynomials: look at the leading term (highest degree). Negative odd power (-x, -x³), left goes up, right goes down. This creates the characteristic 'S-shape' of cubic functions, just flipped when negative!

Question 5

A company's weekly profit (in thousands of dollars) depends on the number of ads $x$ it runs and is modeled by $P(x)=-2(x-3)^2+18$ , for $0\le x\le 6$ . On what interval is $P(x)$ increasing?

$(3,6)$
$[0,3]$ (correct answer)
$[0,6]$
$(0,3)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For this quadratic function P(x) = -2(x-3)^2 + 18, which is a downward-opening parabola with vertex at x=3, the function increases to the left of the vertex (from x=0 to x=3) and decreases to the right (from x=3 to x=6), so the increasing interval within the domain is [0,3]. Choice B correctly identifies the increasing interval as [0,3] because before the vertex at x=3, as x increases, P(x) rises from P(0)=0 to P(3)=18. Choice C gives the y-values instead of the x-values for the interval: [0,6] might seem like the range of outputs, but when we say 'increasing on [0,3],' we mean 'for x-values from 0 to 3, the function increases.' The interval describes the input values (x), not the output values (y). This is a super common confusion! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [2, 5],' we mean 'as x goes from 2 to 5, y is rising.' Similarly, 'positive on (-3, 2)' means 'for x between -3 and 2, y > 0.' The interval describes the horizontal extent, not the vertical range.

Question 6

A piecewise function models the elevation (in meters) of a hiker along a trail:

2x & \text{for } 0\le x\le 4,\\ - x+12 & \text{for } 4< x\le 10. \end{cases}

On what interval is the elevation decreasing?

$[0,4]$
$(4,10]$ (correct answer)
$[0,10]$
$(0,4)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For this piecewise function, let's analyze each piece: For 0 ≤ x ≤ 4, E(x) = 2x, which has positive slope 2, so it's increasing. For 4 < x ≤ 10, E(x) = -x + 12, which has negative slope -1, so it's decreasing. Therefore, the elevation increases from x = 0 to x = 4 (going uphill), then decreases from x = 4 to x = 10 (going downhill). Choice B correctly identifies (4, 10] as the interval where elevation is decreasing because on this portion of the trail, the function E(x) = -x + 12 has a negative slope, meaning the hiker is descending. Choice A gives [0, 4], but this is actually where the function is increasing—the hiker is climbing upward with E(x) = 2x on this interval. For piecewise functions: analyze each piece separately! Look at the formula for each interval and determine if it's increasing (positive slope for linear) or decreasing (negative slope). The behavior can change at the breakpoints!

Question 7

A company's profit (in thousands of dollars) from selling $x$ hundred items is modeled by $P(x)=-x^2+6x-5$ . On what interval is the profit function increasing?

$(3,\infty)$
$(-\infty,3)$ (correct answer)
$(-\infty,0)$
$(-\infty,6)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. To find where the function is increasing/decreasing: looking at the graph from left to right, identify where it goes up/down—those x-values form the interval. For this graph: the quadratic opens downward with vertex at x=3, so increasing on (-∞,3) and decreasing on (3,∞). Choice B correctly identifies the increasing interval as (-∞,3) because before the vertex at x=3, the function rises as x increases. Choice A confuses increasing with decreasing: the function is actually decreasing on (3,∞) because as we move left to right after x=3, the graph goes down. It's easy to mix these up—always read the graph from left to right to determine which is which! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [2, 5],' we mean 'as x goes from 2 to 5, y is rising.' Similarly, 'positive on (-3, 2)' means 'for x between -3 and 2, y > 0.' The interval describes the horizontal extent, not the vertical range. Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!

Question 8

A company's weekly profit (in dollars) from selling $x$ items is modeled by $P(x)=-(x-4)^2+9$ . On what interval is the profit function increasing?

$(4,\infty)$
$(-\infty,4)$ (correct answer)
$(-\infty,\infty)$
$(-\infty,0)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the profit function P(x) = -(x-4)² + 9, this is a parabola in vertex form with vertex at (4, 9). Since the coefficient of the squared term is negative (-1), the parabola opens downward. This means the function increases as we approach the vertex from the left and decreases as we move away from the vertex to the right. Therefore, the function is increasing for all x-values less than 4, which we write as the interval (-∞, 4). Choice B correctly identifies the increasing interval as (-∞, 4) because for a downward-opening parabola, the function rises from the left until it reaches its maximum at x = 4. Choice A gives (4, ∞), which is actually where the function is decreasing—after reaching the maximum at x = 4, the profit drops as we sell more items, perhaps due to oversupply or increased costs. It's easy to confuse which side is increasing! For parabolas, remember: if it opens down (negative coefficient), it increases on the left of the vertex and decreases on the right. If it opens up (positive coefficient), it decreases on the left and increases on the right. The vertex is always the turning point where the behavior changes!

Question 9

A bacteria culture's population is modeled by $N(t)=200\cdot(1.5)^t$ , where $t$ is time in hours. Describe the end behavior as $t\to\infty$ .

As $t\to\infty$ , $N(t)\to 200$
As $t\to\infty$ , $N(t)\to 0$
As $t\to\infty$ , $N(t)\to \infty$ (correct answer)
As $t\to\infty$ , $N(t)\to -\infty$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For end behavior of this exponential function N(t) = 200 · (1.5)^t, since the base 1.5 > 1, as t increases to infinity, N(t) grows without bound to infinity, modeling unlimited bacteria growth over time. Choice D correctly describes the end behavior as t → ∞, N(t) → ∞ because exponential growth with base >1 keeps multiplying and gets larger. Choice C misreads the growth: claiming N(t) → -∞ might confuse with decay, but since base >1, it's growth, not decay to negative—check the base! End behavior shortcut for polynomials: look at the leading term (highest degree). If it's positive even power (x², x⁴), both ends go up. Negative even power (-x², -x⁴), both ends go down. Positive odd power (x, x³), left goes down, right goes up. Negative odd power (-x, -x³), left goes up, right goes down. The leading term dominates for large |x|!

Question 10

The profit (in dollars) from selling $x$ items is modeled by $P(x)=-2x^2+24x-40$ for $0 \leq x \leq 20$ . On what interval is $P(x)$ increasing?

$(0, 20)$
$(-\infty, 6)$
$(0, 6)$ (correct answer)
$(6, 20)$

P(x) = -2x^2 + 24x - 40

, we need to find where it's increasing. Since this is a downward-opening parabola (negative coefficient of

x^2

), it increases from the left boundary up to its vertex, then decreases. The vertex occurs at

x = -\frac{b}{2a} = -\frac{24}{2 \times (-2)} = 6

. Since the domain is

0 \leq x \leq 20

, the function increases from x = 0 to x = 6. Choice B correctly identifies the interval as (0, 6) because the profit rises as we go from selling 0 items to selling 6 items, after which it starts to decrease. Choice D gives (0, 20), which would mean the function increases over the entire domain—but this misses that after x = 6, the profit actually decreases as selling too many items reduces profit! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [0, 6],' we mean 'as x goes from 0 to 6, y is rising.' The interval describes the horizontal extent, not the vertical range.

Question 11

A company's weekly profit (in dollars) from selling $x$ items is modeled by $P(x)=-x^2+8x-12.$ On what interval is the profit function increasing?

$(4,\infty)$
$(-\infty,4)$ (correct answer)
$(-\infty,-4)$
$(0,\infty)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the profit function P(x) = -x² + 8x - 12, this is a downward-opening parabola (negative x² coefficient), which increases from the left up to its vertex, then decreases afterward. The vertex occurs at x = -b/(2a) = -8/(2(-1)) = 4, so the function increases on (-∞, 4) and decreases on (4, ∞). Choice B correctly identifies the increasing interval as (-∞, 4) because for a downward-opening parabola, the function rises from negative infinity up to the vertex at x = 4. Choice A gives (4, ∞), which is actually where the function is decreasing—after the peak at x = 4, the profit starts falling as we produce too many items. To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on (-∞, 4),' we mean 'as x goes from negative infinity to 4, y is rising.' For parabolas, remember: if it opens down, it increases then decreases; if it opens up, it decreases then increases.

Question 12

A company's profit function is $P(x) = -2x^2 + 40x - 150$ where $x$ is the number of items produced (in hundreds) and $P(x)$ is profit in thousands of dollars. For what range of production levels does the company earn a positive profit?

The company earns positive profit when producing more than 10 hundred items
The company earns positive profit when producing between 5 and 15 hundred items
The company earns positive profit when producing between 500 and 1500 items (correct answer)
The company earns positive profit when producing fewer than 20 hundred items

Explanation: When you encounter a profit function and need to find when profit is positive, you're solving a quadratic inequality. The key is finding where

P(x) > 0

, which means determining where the parabola is above the x-axis. To find when

P(x) = -2x^2 + 40x - 150 > 0

, first find where the profit equals zero by solving

-2x^2 + 40x - 150 = 0

. Dividing by -2 gives us

x^2 - 20x + 75 = 0

. This factors as

(x - 5)(x - 15) = 0

, so the roots are

x = 5

and

x = 15

. Since the coefficient of

x^2

is negative (-2), this parabola opens downward, meaning profit is positive between the roots. Therefore, profit is positive when

5 < x < 15

. Remember that

x

represents hundreds of items, so this means between 500 and 1500 items, making C correct. Choice A suggests profit is positive only above 1000 items, missing the upper bound where profit becomes negative again. Choice B correctly identifies the range in hundreds but fails to convert back to actual items produced. Choice D claims profit is positive below 2000 items, ignoring that profit is also negative below 500 items. The key strategy here is remembering that quadratic inequalities with downward-opening parabolas are positive between their roots, and always check your units carefully. Many algebra problems involve converting between different scales (hundreds vs. actual quantities), so read the variable definitions closely.

Question 13

A water balloon is launched upward from a platform. The height $h(t) = -16t^2 + 64t + 80$ gives the balloon's height in feet after $t$ seconds. Which statement correctly describes a key feature of this function in the context of the problem?

The balloon reaches its maximum height of 144 feet at $t = 2$ seconds after launch (correct answer)
The balloon hits the ground exactly 6 seconds after being launched from the platform
The balloon's height is decreasing most rapidly during the first second after launch
The balloon returns to its initial launch height after exactly 4 seconds of flight

Explanation: To find the maximum height, we complete the square or use the vertex formula. For

h(t) = -16t^2 + 64t + 80

, the vertex occurs at

t = -\frac{64}{2(-16)} = 2

. At

t = 2

h(2) = -16(4) + 64(2) + 80 = -64 + 128 + 80 = 144

feet. Choice B is wrong because solving

-16t^2 + 64t + 80 = 0

gives

t = 5

(not 6). Choice C is wrong because the balloon is increasing in height during the first 2 seconds. Choice D is wrong because the initial height is 80 feet, and

h(4) = 80

, so it returns to launch height at

t = 4

, but this takes 4 seconds, not exactly 4 seconds as a special feature.

Question 14

A function $g(x)$ has the following properties: $g(0) = -3$ , $g(2) = 0$ , $g(4) = 0$ , and $g(6) = -3$ . The function is symmetric about the line $x = 3$ . Based on this information, which sketch best represents the key features that $g(x)$ must have?

A parabola opening upward with vertex at $(3, 1)$ and x-intercepts at $x = 2$ and $x = 4$
A parabola opening downward with vertex at $(3, 1)$ and x-intercepts at $x = 2$ and $x = 4$ (correct answer)
A parabola opening upward with vertex at $(3, -4)$ and x-intercepts at $x = 1$ and $x = 5$
A parabola opening downward with vertex at $(3, -4)$ and x-intercepts at $x = 1$ and $x = 5$

Explanation: Given the symmetry about

x = 3

and the points provided:

g(0) = g(6) = -3

(symmetric about

x = 3

), and

g(2) = g(4) = 0

(x-intercepts). Since the function has value

-3

at the symmetric points

x = 0

and

x = 6

, and

0

x = 2

and

x = 4

, the vertex at

x = 3

must be above the x-intercepts. The parabola opens downward with vertex at

(3, 1)

. Choice A has correct vertex and intercepts but wrong orientation. Choice C has wrong vertex location and intercepts. Choice D has wrong vertex and intercepts despite correct downward orientation.

Question 15

A cyclist's distance from home during a 6-hour bike ride is recorded every hour. The cyclist starts at home, rides to a park 15 miles away (arriving after 2 hours), stays at the park for 1 hour, then rides to a friend's house 25 miles from home (arriving after 5 hours), and remains there.

Based on this scenario, during which time interval is the distance function decreasing at the fastest rate?

From hour 0 to hour 2, because this represents the longest continuous period of travel
From hour 3 to hour 5, because the cyclist travels the greatest total distance during this period
From hour 2 to hour 3, because the function remains constant at 15 miles throughout this interval
The distance function is never decreasing during this 6-hour period based on the given information (correct answer)

Explanation: Analyzing each interval: Hours 0-2: distance increases from 0 to 15 miles (increasing). Hours 2-3: distance stays at 15 miles (constant). Hours 3-5: distance increases from 15 to 25 miles (increasing). Hours 5-6: distance stays at 25 miles (constant). The distance from home never decreases because the cyclist never moves closer to home than their current position. Choice A describes an increasing interval. Choice B describes another increasing interval. Choice C describes a constant interval, not decreasing.

Question 16

A spring's displacement from equilibrium (in centimeters) is modeled by $s(t)=2\sin\left(\frac{\pi}{4}t\right)$ , where $t$ is time in seconds. On what interval from $t=0$ to $t=8$ is the displacement positive?

$(0,8)$
$(0,4)$ (correct answer)
$(0,2)\cup(4,6)$
$(2,4)\cup(6,8)$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. To find where the function is positive/negative: identify where the graph is above/below the x-axis—those x-values form the interval. For this graph: with period 8, sine is positive from t=0+ to t=4 (excluding endpoints where zero). Choice A correctly identifies the positive interval as (0,4) because that's where sin(π/4 t) >0 in [0,8]. Choice B describes the wrong interval: it says positive on (0,2)∪(4,6), but checking where the graph is above the x-axis, we see it's actually on (0,4) continuously—always verify by checking specific points in your interval! To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on [2, 5],' we mean 'as x goes from 2 to 5, y is rising.' Similarly, 'positive on (-3, 2)' means 'for x between -3 and 2, y > 0.' The interval describes the horizontal extent, not the vertical range.

Question 17

A function is described as follows: It starts at the point $(0,2)$ , increases to a maximum at $(3,5)$ , then decreases and crosses the $x$ -axis at $(6,0)$ . Where does the function change from increasing to decreasing?

$x=3$ (correct answer)
$x=6$
$(0,2)$
$x=0$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. A maximum is the highest point on a graph (or on some portion of it), giving the largest y-value, while a minimum is the lowest point, giving the smallest y-value. For a parabola that opens up, the vertex is the minimum; if it opens down, the vertex is the maximum. In real-world problems, these tell you the best or worst outcome! To sketch a function with starting at (0,2), maximum at (3,5), x-intercept at (6,0), we: (1) Plot key points like intercepts and extrema: (0,2), (3,5), (6,0), (2) Note behavior: increasing on (0,3), decreasing on (3,6), (3) Connect with appropriate curve type, (4) Verify sketch shows all required features. The sketch doesn't need to be perfect, just show the main features clearly! Choice D correctly identifies the change from increasing to decreasing at x=3 because that's the maximum point where the behavior switches. Choice B confuses with the x-intercept: x=6 is where it crosses x-axis, but the switch happens at the peak—look for where the description says 'increases to a maximum' then 'decreases'! For sketching from verbal descriptions: (1) Make a checklist of all required features, (2) Plot any specific points given (intercepts, extrema), (3) Identify regions where function increases, decreases, is positive, is negative, (4) Connect the dots with the right curve type (line, parabola, etc.) making sure all features are visible. Don't worry about making it perfect—as long as the key features are clearly shown, you're good!

Question 18

Describe the end behavior of $f(x)=-2x^3+5x^2$ .

As $x\to\infty$ , $f(x)\to\infty$ and as $x\to-\infty$ , $f(x)\to-\infty$
As $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to\infty$ (correct answer)
As $x\to\infty$ , $f(x)\to\infty$ and as $x\to-\infty$ , $f(x)\to\infty$
As $x\to\infty$ , $f(x)\to-\infty$ and as $x\to-\infty$ , $f(x)\to-\infty$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. Looking at the graph: end behavior dominated by -2x³ term, so as x→∞, f(x)→-∞; as x→-∞, f(x)→∞. Choice B correctly describes the end behavior as x→∞, f(x)→-∞ and x→-∞, f(x)→∞ because for negative leading coefficient on odd degree, right end down, left end up. Choice A reverses the directions: it's the opposite for negative odd degree—always check the sign and parity of the leading term! End behavior shortcut for polynomials: look at the leading term (highest degree). If it's positive even power (x², x⁴), both ends go up. Negative even power (-x², -x⁴), both ends go down. Positive odd power (x, x³), left goes down, right goes up. Negative odd power (-x, -x³), left goes up, right goes down. The leading term dominates for large |x|!

Question 19

A ride at an amusement park moves up and down in a repeating pattern. The height (in meters) above the ground is modeled by $h(t)=12+4\sin\left(\frac{\pi}{3}t\right)$ , where $t$ is time in seconds. What is the period of $h(t)$ ?

$3$
$6$ (correct answer)
$2\pi$
$12$

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For the sinusoidal function h(t) = 12 + 4sin(πt/3), the period is determined by the coefficient of t inside the sine function. The general form is A·sin(Bt) + C, where the period = 2π/B. Here, B = π/3, so period = 2π ÷ (π/3) = 2π × (3/π) = 6 seconds. This means the ride completes one full up-and-down cycle every 6 seconds. Choice B correctly identifies the period as 6 because dividing 2π by the coefficient π/3 gives us 6 seconds for one complete cycle. Choice C gives 2π, which might come from confusing the period formula—remember to divide 2π by the coefficient of t, not just use 2π itself! For sine and cosine functions: Period = 2π/B where B is the coefficient of the variable inside the function. A larger B means faster oscillation (shorter period), while smaller B means slower oscillation (longer period). The ride repeats its pattern every 6 seconds!

Question 20

A ball is tossed upward from a platform. Its height above the ground (in meters) after $t$ seconds is modeled by the function $h(t)=-5t^2+20t+2$ for $t\ge 0$ . What does the maximum of $h(t)$ represent in this context?

The constant rate at which the ball rises
The greatest height the ball reaches above the ground (correct answer)
The time when the ball hits the ground
The height of the ball at $t=0$ seconds

Explanation: This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. A maximum is the highest point on a graph (or on some portion of it), giving the largest y-value, while a minimum is the lowest point, giving the smallest y-value. For a parabola that opens up, the vertex is the minimum; if it opens down, the vertex is the maximum. In real-world problems, these tell you the best or worst outcome! In this context where the function models the height of a ball over time, the maximum of h(t) = -5t² + 20t + 2 represents the highest point the ball reaches during its flight. Since the coefficient of t² is negative (-5), the parabola opens downward, so the vertex is indeed a maximum. To find it, we use t = -b/(2a) = -20/(2×(-5)) = 2 seconds, giving h(2) = -5(4) + 20(2) + 2 = 22 meters. Choice B correctly interprets the maximum as 'The greatest height the ball reaches above the ground' because the maximum y-value (22 meters) represents the peak height during the ball's trajectory. Choice A confuses the y-intercept (initial height at t=0, which is 2 meters) with the maximum—the ball starts at 2 meters but goes much higher! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!