When you encounter questions about identifying function types, focus on the defining characteristics that make each type unique. A linear function must produce a straight line when graphed and can only contain variables raised to the first power.

The equation $$f(x) = 3x^2 - 2x + 5$$ is actually a quadratic function because it contains $$x^2$$, meaning the variable is raised to the second power. When you graph this equation, it creates a parabola (a U-shaped curve), not a straight line. Aaliyah's error is thinking that substituting one specific value (like $$x = 1$$) and getting a result that fits the linear form $$y = mx + b$$ means the entire function is linear. But linear functions must produce straight lines for ALL possible values of $$x$$, not just one point.

Looking at the wrong answers: Choice A incorrectly focuses on the number of terms rather than the powers of variables—linear functions can have multiple terms as long as variables aren't raised above the first power. Choice B is wrong because linear functions can absolutely have positive coefficients; the sign of coefficients doesn't determine function type. Choice D incorrectly suggests that the size of the constant term matters for linearity, but constants of any size are perfectly acceptable in linear functions.

Remember this key distinction: linear functions contain only first-power variables and graph as straight lines, while quadratic functions contain squared variables and graph as parabolas. Don't let one data point fool you into thinking the entire function behaves linearly.

Function B passes through (0,7) and (4,19). Using the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19 - 7}{4 - 0} = \frac{12}{4} = 3$$. We can verify: Function A gives $$y = 2(4) + 11 = 19$$ at $$x = 4$$, confirming the intersection. Choice A uses incorrect reasoning about y-intercept differences. Choice B incorrectly uses the x-coordinate. Choice C makes an arithmetic error in the division.

The corrected equation is $$y = -3x + 8$$ (same slope of -3, but y-intercept changed to 8). At $$x = 2$$ minutes: $$y = -3(2) + 8 = -6 + 8 = 2$$. So the point is (2, 2). Choice B incorrectly applies the y-intercept change to the point rather than recalculating. Choice C incorrectly suggests slope adjustment. Choice D misapplies the original equation and adds values incorrectly.

In the linear function $$B = 45m + 120$$, the slope 45 represents the monthly deposit amount, and 120 is the initial balance (y-intercept). Changing the monthly deposit to $60 changes the slope from 45 to 60, making the graph steeper, but the initial balance remains $120. Choice A incorrectly changes the y-intercept. Choice C confuses slope change with vertical shift. Choice D incorrectly describes the slope change and adds an irrelevant horizontal shift.

When you see a problem about something changing at a constant rate over time, you're dealing with a linear relationship. The key is identifying the rate of change and the starting value to write the equation in the form $$y = mx + b$$.

First, let's find the rate at which the tank drains. The tank goes from 500 gallons to 350 gallons in 3 hours, so it loses $$500 - 350 = 150$$ gallons total. Since this happens over 3 hours, the rate is $$150 ÷ 3 = 50$$ gallons per hour. Because the tank is draining (losing water), this rate is negative: $$-50$$ gallons per hour.

The equation follows the pattern $$G = (\text{rate}) \times t + (\text{starting amount})$$, which gives us $$G = -50t + 500$$. This matches choice C.

Choice A incorrectly shows the tank gaining 50 gallons per hour with a positive rate, but the tank is draining. Choice B uses the correct starting value and negative direction, but mistakenly uses the total loss (150 gallons) as the hourly rate instead of dividing by 3 hours. Choice D completely misunderstands the problem by treating 350 as the starting point and showing the tank filling rather than draining.

Study tip: For constant rate problems, always calculate the rate by finding the total change and dividing by the time interval. Pay careful attention to whether the quantity is increasing (positive rate) or decreasing (negative rate).

To determine if a function is linear, you must verify it produces a straight line for ALL input values, not just one. The equation $$P = 100 \cdot 2^t$$ is exponential (due to the variable in the exponent), giving points like (0,100), (1,200), (2,400), (3,800) which don't form a straight line. Testing only one point is insufficient. Choice B makes a false claim about coefficient limits. Choice C incorrectly focuses on variable count. Choice D attempts an irrelevant calculation correction.

This question tests interpreting y=mx+b as defining a linear function (straight-line graph, constant slope m, y-intercept b) and distinguishing from non-linear functions (curved graphs, variable squared/in denominator/in absolute value/as exponent), specifically in a real-world context. Linear function y=mx+b: m is slope (rate of change: y increases by m per unit x), b is y-intercept (initial value when x=0, where line crosses y-axis), graph is straight line through (0,b) with constant slope m. Non-linear functions have x with exponent ≠1, or in denominator, or in other function: y=x² graphs as parabola (curved, points (1,1),(2,4),(3,9) not on straight line), y=1/x graphs as hyperbola (curved), y=|x| is V-shape (not single straight line)—all have non-constant slopes (curvature indicates slope varies). For example, y=3x+2 is linear with a straight graph where m=3 means y rises by 3 per x, b=2 is the start, versus y=x² with points showing varying increases like from 1 to 4 then to 9. The correct choice is B, where m=2 is the cost per snack (rate) and b=6 is the ticket fee (fixed cost when x=0), matching the model's interpretation. A common error is interpreting m and b backwards, like thinking the fixed fee is the slope, or confusing them with non-cost elements like starting snacks. To identify and interpret: (1) check form y=mx+b for linearity, (2) confirm x to power 1, (3) graph if needed for straight line, (4) calculate constant slope; here, m is the per-unit rate ($2/snack), b is initial value ($6 ticket), with mistakes like ignoring context or claiming non-linear due to positive values.

This question tests interpreting y=mx+b as defining a linear function with a straight-line graph and constant slope m, while identifying non-linear functions with curved graphs or non-constant rates, such as those with x squared, in denominators, absolute values, or exponents. Linear functions like y=mx+b have constant slope m (y changes by m per unit x) and y-intercept b, graphing straight through (0,b); non-linear ones vary, e.g., y=x² is a parabola with increasing slope (points (1,1),(2,4),(3,9) show y-differences 3,5—not constant), y=1/x hyperbola, y=|x| V-shape with slope shift. For instance, y=3x+2 is linear with constant rate 3 and straight graph, but y=x² curves with points not collinear, demonstrating variable rate (slope between (0,0)-(1,1) is 1, but (1,1)-(2,4) is 3). The non-linear function here is C, y=x², lacking constant rate and straight graph, while A, B, D are linear in y=mx+b form. Common errors include calling y=x² linear because it has x (ignoring exponent 2 causing curvature) or misinterpreting m and b in linear ones, like swapping slope and intercept. Identifying linear: (1) rewrite as y=mx+b, (2) check x exponent=1, (3) graph for straightness, (4) confirm constant slope via points. Interpreting: m is rate/steepness, b starting value; mistakes: assuming all variable equations are linear or claiming curved graphs have constant rates.

This question tests interpreting y=mx+b as defining a linear function with straight-line graph, constant slope m from points, and y-intercept b, distinguishing from non-linear not fitting straight lines. Linear y=mx+b: m=(y2-y1)/(x2-x1) constant, b=y when x=0; non-linear like y=x² don't have constant m between points, curving instead of straight. For example, points (0,2),(1,5) give m=3, y=3x+2 linear straight; versus (0,0),(1,1),(2,4) for y=x² with varying m=1 then 3, curved. Here, points (0,-3),(2,1) give m=(1-(-3))/(2-0)=2, b=-3, so y=2x-3 (A) matches the line. Common errors: wrong m like -2 (B) or 1/2 (C), or sign flip to +3 (D), miscalculating slope or intercept. Identifying: (1) compute m from points, (2) find b at x=0, (3) write y=mx+b, (4) verify other points. Interpreting: m rate between points, b start; mistakes: swapping signs or confusing rise/run.

This question tests interpreting y=mx+b as defining a linear function (straight-line graph, constant slope m, y-intercept b) and distinguishing from non-linear functions (curved graphs, variable squared/in denominator/in absolute value/as exponent). Linear function y=mx+b: m is slope (rate of change: y increases by m per unit x), b is y-intercept (initial value when x=0, where line crosses y-axis), graph is straight line through (0,b) with constant slope m. Non-linear functions have x with exponent ≠1, or in denominator, or in other function: y=x² graphs as parabola (curved, points (1,1),(2,4),(3,9) not on straight line), y=1/x graphs as hyperbola (curved), y=|x| is V-shape (not single straight line)—all have non-constant slopes (curvature indicates slope varies). For example, y=3x+2 is linear with a straight graph passing through (0,2) and rising 3 units per 1 unit right, whereas y=x² has points like (0,0), (1,1), (2,4) demonstrating curvature as the slope increases. The correct choice is C, explaining that y=|x| is not linear because it forms a V-shape, not a single straight line, and cannot be expressed as one y=mx+b equation for all x. A common error is thinking y=|x| is linear due to its straight pieces, but the change in slope direction makes it non-linear overall. Identifying linear: (1) check form (can write as y=mx+b? yes→linear), (2) check exponent (x¹ only? yes→linear, x², x⁰, x⁻¹ etc.→non-linear), (3) check graph (straight→linear, curved→non-linear), (4) verify constant slope (calculate between multiple point pairs, same→linear, varies→non-linear). Interpreting: m in y=mx+b is slope (rise/run, rate, steepness), b is y-intercept (where x=0, starting value), context meaning (y=3x+20 for cost: 3=$/item, 20=fixed cost). Mistakes: thinking any equation with x,y is linear (ignoring exponents, denominators), confusing m and b roles, claiming curved graphs are linear.