SAT Math Quiz: Linear Functions

What this quiz covers

This quiz focuses on Linear Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

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.

Question 1

1. $-\frac{7}{4}$
2. $4$ (correct answer)
3. $\frac{7}{4}$
4. $-4$

Explanation: We need to find x when g(x) = -7, where g(x) = -4x + 9. Set up the equation: -4x + 9 = -7. Solve for x by first subtracting 9 from both sides: -4x = -7 - 9 = -16. Then divide both sides by -4: x = -16/(-4) = 4. We can verify: g(4) = -4(4) + 9 = -16 + 9 = -7 ✓. A common error is sign confusion when dividing by negative numbers. Remember that a negative divided by a negative gives a positive result.

Question 2

1. $14$ (correct answer)
2. $26$
3. $-26$
4. $8$

Explanation: We need to evaluate f(8) where f(x) = 2.5x - 6. To find f(8), substitute 8 for x in the function: f(8) = 2.5(8) - 6. Calculate step by step: 2.5 × 8 = 20, then 20 - 6 = 14. Therefore, f(8) = 14. A common error is arithmetic mistakes when multiplying decimals; converting 2.5 to 5/2 can help avoid this. When evaluating functions, always substitute carefully and perform operations in the correct order.

Question 3

1. $y+7=-\frac{3}{2}(x-4)$
2. $y-7=-\frac{3}{2}(x-4)$ (correct answer)
3. $y-4=-\frac{3}{2}(x-7)$
4. $y-7=\frac{3}{2}(x-4)$

Explanation: The question asks for point-slope form of a line with slope -3/2 passing through (4, 7). Point-slope form is y - y1 = m(x - x1), so substituting m = -3/2, x1 = 4, and y1 = 7 gives y - 7 = -3/2(x - 4), the equation that subtracts 7 on the left and 4 inside the parentheses while keeping the negative coefficient. The version written with y + 7 mishandles the sign, since subtracting a positive 7 yields y - 7; that form would correspond to the point (4, -7). The version with y - 4 and (x - 7) swaps the coordinates, dropping the x-value into the y-slot. And the version with a positive 3/2 out front loses the negative sign, describing a line that rises rather than falls.

Question 4

1. When $x$ increases by 4, $y$ increases by 1
2. When $x$ increases by 1, $y$ increases by 3
3. When $x$ increases by 3, $y$ decreases by 1
4. When $x$ increases by 3, $y$ increases by 1 (correct answer)

Explanation: In y = (1/3)x + 4 the slope is the coefficient 1/3, and slope means change in y divided by change in x. So a run of 3 produces a rise of 3 times 1/3, which equals 1. That is exactly the statement that when x increases by 3, y increases by 1. The statement that y increases by 3 when x increases by 1 flips the fraction over, treating the slope as 3 rather than 1/3. The statement that y increases by 1 when x increases by 4 pulls the 4 from the y-intercept; the intercept tells you where the line sits, not how steeply it climbs. The statement that y decreases by 1 when x increases by 3 gets the size right but the direction wrong, since a positive slope means y rises as x rises.

Question 5

1. $18
2. $22
3. $20 (correct answer)
4. $16

Explanation: Let b be the base pay and c be the commission per item. We have two equations: $b + 12c = 460$ and $b + 20c = 620$. Subtracting the first from the second: $8c = 160$, so $c = 20$. The commission per item is $20$. To verify: $b = 460 - 12(20) = 460 - 240 = 220$, and checking: $220 + 20(20) = 220 + 400 = 620$ ✓. A common error is setting up the equations incorrectly or making arithmetic mistakes when solving the system. When dealing with pay structures, clearly define your variables before writing equations.

Question 6

1. 12 (correct answer)
2. 5
3. 17
4. 60

Explanation: When $x=0$, the cost is the y-intercept, 12 dollars. The other values use the slope, add incorrectly, or multiply the slope and intercept.

Question 7

1. 0.36
2. 0.3 (correct answer)
3. 36
4. 0.12

Explanation: The slope is $(131-95)/(220-100) = 36/120 = 0.30 dollars per mile. The other choices come from miscomputing the slope or using incorrect units.

Question 8

1. $y-2=-\frac{4}{3}(x-4)$
2. $y+2=-\frac{3}{4}(x-4)$
3. $y-2=-\frac{3}{4}(x-4)$ (correct answer)
4. $y-4=-\frac{3}{4}(x-2)$

Explanation: The question asks for the point-slope form of a line with slope m = -3/4 passing through the point (4, 2). The point-slope form is y - y1 = m(x - x1), so substituting gives y - 2 = -3/4 (x - 4), which is choice C. This form highlights the slope and a specific point on the line. A common computational error is inverting the fraction in the slope, like using -4/3 instead of -3/4, as in choice A. Another mistake is swapping the x and y coordinates in the formula, leading to something like y - 4 = -3/4 (x - 2) as in choice D. To verify, you can convert to slope-intercept form: y = -3/4 x + 3 + 2 = -3/4 x + 5, and check the point satisfies it. A strategy for these problems is to remember point-slope is useful when you know a point and slope directly.

Question 9

1. $-\frac{1}{2}$
2. $4$ (correct answer)
3. $-4$
4. $-\frac{7}{4}$

Explanation: We need to solve for $x$ when $f(x) = -7$ given $f(x) = -4x + 9$. Setting up the equation: $-7 = -4x + 9$. Subtracting 9 from both sides: $-16 = -4x$. Dividing by -4: $x = 4$. We can verify: $f(4) = -4(4) + 9 = -16 + 9 = -7$ ✓. Common errors include sign mistakes when solving or forgetting to divide by the coefficient of $x$. When solving linear equations, always verify your answer by substituting back into the original function.

Question 10

1. $y-1=-\frac{3}{2}(x-4)$ (correct answer)
2. $y-4=-\frac{3}{2}(x-1)$
3. $y+1=-\frac{3}{2}(x-4)$
4. $y-1=\frac{3}{2}(x-4)$

Explanation: We need to write the equation of a line with slope $-\frac{3}{2}$ passing through $(4,1)$ in point-slope form. Point-slope form is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Substituting our values: $y - 1 = -\frac{3}{2}(x - 4)$. Common errors include using the wrong signs or confusing point-slope form with slope-intercept form. Remember that in point-slope form, we subtract the coordinates of the given point, not add them.

Question 11

1. $T=2t+66$
2. $T=-2t+70$ (correct answer)
3. $T=-\frac{1}{2}t+68.5$
4. $T=\frac{1}{2}t+67.5$

Explanation: Temperature T changes linearly with time t, with T = 68 when t = 1 and T = 60 when t = 5. The slope is m = (60 - 68)/(5 - 1) = -8/4 = -2 degrees per hour. Using point-slope form with (1, 68): T - 68 = -2(t - 1), which gives T = -2t + 2 + 68 = -2t + 70. The equation is T = -2t + 70. Common errors include mixing up which variable is dependent (T) versus independent (t), or sign errors in the slope calculation. In physics contexts, negative slope often indicates a decreasing quantity over time.

Question 12

1. $-1$ (correct answer)
2. $1$
3. $-\frac{3}{2}$
4. $-\frac{2}{3}$

Explanation: To find the slope between two points, we use the formula m = (y₂ - y₁)/(x₂ - x₁). Using points (-2, 5) and (4, -1), we calculate: m = (-1 - 5)/(4 - (-2)) = -6/6 = -1. The key is to be consistent with which point is (x₁, y₁) and which is (x₂, y₂), and to be careful with negative signs when subtracting. A common error is forgetting that subtracting a negative becomes addition in the denominator. When you see two points, immediately think slope formula.

Question 13

1. $20$
2. $0$
3. $4$ (correct answer)
4. $-4$

Explanation: We need to find the x-value where g(x) = 0 for the function g(x) = 5x - 20. Setting g(x) = 0: 5x - 20 = 0, so 5x = 20, which gives x = 4. This is the x-intercept of the function. A common error is to find the y-intercept instead (when x = 0, g(0) = -20) or to solve 5x - 20 = x, confusing the function value with the input. To find where a linear function equals zero, set the function equal to zero and solve for x.

Question 14

1. $-3$ (correct answer)
2. $-21$
3. $21$
4. $-13$

Explanation: We need to find the value of a linear function with slope -4 and y-intercept 9 when x = 3. The function is f(x) = -4x + 9. Evaluating at x = 3: f(3) = -4(3) + 9 = -12 + 9 = -3. A common error is to make arithmetic mistakes with negative numbers, such as computing -4(3) as -7 or forgetting the negative sign entirely. When evaluating linear functions, write out each step to avoid sign errors.

Question 15

1. $y=-x+1$ (correct answer)
2. $y=x+1$
3. $y=-x-1$
4. $y=\tfrac{1}{3}x+\tfrac{1}{3}$

Explanation: We need to find the equation of line ℓ passing through (-4,5) and (2,-1). First, calculate the slope: m = (-1 - 5)/(2 - (-4)) = -6/6 = -1. Using point-slope form with point (2,-1): y - (-1) = -1(x - 2), which gives y + 1 = -x + 2, so y = -x + 1. We can verify using the other point: y = -(-4) + 1 = 4 + 1 = 5 ✓. A common error is miscalculating the slope by mixing up the order of subtraction or making sign errors. When finding equations from two points, always verify both points satisfy your final equation.

Question 16

1. $y=3x-2$ (correct answer)
2. $y=-3x-2$
3. $y=\tfrac{1}{3}x-2$
4. $y=3x+2$

Explanation: This question describes a line that crosses the y-axis at -2 (so b = -2) and rises 3 units for every 1 unit to the right (so slope m = 3). Using slope-intercept form y = mx + b, we get y = 3x - 2. The phrase "rises 3 units for every 1 unit to the right" directly gives us the slope as rise/run = 3/1 = 3. A common error is to confuse the y-intercept with a different point or to misinterpret the slope description. When given verbal descriptions of slope, translate "rise" and "run" carefully into the slope ratio.

Question 17

1. The tank drains 8 L each minute
2. The volume after 1 minute
3. The initial volume, 120 L (correct answer)
4. The time when the tank is empty

Explanation: In this context, V(t) represents volume after t minutes, starting with 120 liters and draining at 8 liters per minute, giving us V(t) = 120 - 8t. The y-intercept occurs when t = 0, which gives V(0) = 120 - 8(0) = 120 liters. This represents the initial volume in the tank before any draining begins. Students often confuse the y-intercept with other features like the rate of change (slope) or the x-intercept (when the tank is empty). In real-world linear models, the y-intercept always represents the initial condition—the value when time (or the independent variable) equals zero.

Question 18

1. $4$
2. $6$ (correct answer)
3. $24$
4. $26$

Explanation: This problem involves finding the monthly fee from two total cost data points. Let's denote the sign-up fee as S and monthly fee as M, then after 3 months: S + 3M = 41, and after 7 months: S + 7M = 65. Subtracting the first equation from the second: (S + 7M) - (S + 3M) = 65 - 41, which gives us 4M = 24, so M = 6. The monthly fee is $6. A common error is trying to solve for both unknowns instead of recognizing that subtraction eliminates the sign-up fee. When dealing with linear relationships involving fixed and variable costs, use the difference between two data points to isolate the rate of change.

Question 19

1. $y=4.50x+2.25$
2. $y=2.25x+4.50$ (correct answer)
3. $y=2.25x-4.50$
4. $y=4.50x-2.25$

Explanation: This question asks us to write an equation for a taxi fare that includes a base fee and a per-mile charge. In the slope-intercept form y = mx + b, the slope m represents the rate of change (cost per mile) and the y-intercept b represents the initial value (base fee). Here, the base fee is $4.50 (the y-intercept) and the cost per mile is $2.25 (the slope), so the equation is y = 2.25x + 4.50. A common error is confusing which value is the slope and which is the y-intercept—remember that the base fee is what you pay even for 0 miles, making it the y-intercept. When dealing with word problems, identify the fixed cost (y-intercept) and the variable rate (slope) before writing the equation.

Question 20

1. $y-8=\frac{1}{3}(x-3)$ (correct answer)
2. $y-3=\frac{1}{3}(x-8)$
3. $y-8=3(x-3)$
4. $y+8=\frac{1}{3}(x+3)$

Explanation: The question asks for the point-slope form of a line through (3,8) with slope 1/3. The form is y - y1 = m(x - x1), so y - 8 = (1/3)(x - 3). This directly uses the given point and slope. It models the linear equation without expanding. A common error is switching x and y coordinates, like y-3=(1/3)(x-8). Another mistake might be using reciprocal slope like 3. For point-slope, plug in the known point and slope carefully.