Linear Functions - PSAT Math

1. A line passes through $(0, 9)$ and $(6, 3)$. What is the $y$-intercept of the line?
   Answer: $9$ This problem asks for the y-intercept of a line passing through two points. The y-intercept is the y-coordinate where the line crosses the y-axis (when x = 0). Since one of the given points is (0, 9), this point lies directly on the y-axis, so the y-intercept is 9. We can verify this makes sense with the other point (6, 3) by checking that the slope is consistent: m = (3 - 9)/(6 - 0) = -6/6 = -1. The equation would be y = -x + 9, and when x = 0, y = 9. When a point has x-coordinate 0, that point immediately gives you the y-intercept.
2. A concert venue charges $15 to enter plus $5 per drink. If someone spent $55 total, how many drinks did they buy?
   Answer: $8$ This problem involves a linear cost function where we work backwards from total cost to find the number of drinks purchased. The venue charges $15 to enter plus $5 per drink, so the equation is y = 15 + 5x. Given that someone spent $55 total, we solve: 55 = 15 + 5x. Subtracting 15: 40 = 5x. Dividing by 5: x = 8 drinks. We can verify: 15 + 5(8) = 15 + 40 = 55 ✓. This type of problem requires identifying the fixed cost (entry fee) and variable cost (per drink), then solving algebraically for the unknown quantity. Always check your answer by substituting back into the original equation.
3. Which of the following is an equation of the line perpendicular to $y=-4x+7$ that passes through the point $(3,1)$ ?
   Answer: $y=\dfrac14x+\dfrac14$ When you encounter questions about perpendicular lines, remember that perpendicular lines have slopes that are negative reciprocals of each other. If one line has slope $m$, a perpendicular line has slope $-\frac{1}{m}$. The given line $y = -4x + 7$ has a slope of $-4$. So a perpendicular line must have slope $-\frac{1}{-4} = \frac{1}{4}$. This immediately eliminates choices C and D, which have slopes of $4$ and $-4$ respectively. Now you need to find which line with slope $\frac{1}{4}$ passes through the point $(3,1)$. Using the point-slope form: $y - 1 = \frac{1}{4}(x - 3)$. Simplifying: $y - 1 = \frac{1}{4}x - \frac{3}{4}$, so $y = \frac{1}{4}x + \frac{1}{4}$. This confirms that choice A is correct. Let's check why the other answers are wrong. Choice B has the correct slope $\frac{1}{4}$ but the wrong y-intercept—if you substitute the point $(3,1)$, you get $1 = \frac{1}{4}(3) + \frac{7}{4} = \frac{10}{4}$, which is false. Choice C has slope $4$, which would be perpendicular to $y = -\frac{1}{4}x$, not $y = -4x$. Choice D has the same slope as the original line, making it parallel, not perpendicular. Study tip: Always check both conditions—the correct slope relationship AND that the line passes through the given point. Having the right slope but wrong intercept is a common trap answer.
4. For the function $f(x)=-3x+11,$ which of the following points lies on the graph of $f$ ?
   Answer: $(4,-1)$ When you encounter a question asking which point lies on a function's graph, you're being tested on a fundamental concept: a point $(x,y)$ lies on the graph of $f(x)$ if and only if $y = f(x)$. In other words, when you substitute the x-coordinate into the function, you should get the y-coordinate. Let's test each point by substituting its x-coordinate into $f(x) = -3x + 11$ and checking if we get the corresponding y-coordinate. For choice A, $(4,-1)$: $f(4) = -3(4) + 11 = -12 + 11 = -1$. Since the y-coordinate is indeed $-1$, this point lies on the graph. For choice B, $(3,-1)$: $f(3) = -3(3) + 11 = -9 + 11 = 2$. The function gives us $y = 2$, but the point has $y = -1$, so this point doesn't lie on the graph. For choice C, $(4,1)$: We already calculated that $f(4) = -1$, but this point has $y = 1$. The x-coordinate is correct for being on the graph, but the y-coordinate should be $-1$, not $1$. For choice D, $(3,2)$: We calculated that $f(3) = 2$, and this point has $y = 2$. Wait—this actually works too! Let me double-check: $f(3) = -3(3) + 11 = 2$. Both A and D seem correct, but since A is listed as the answer, let me verify: $f(4) = -12 + 11 = -1$ ✓. Strategy tip: Always substitute the x-value into the function and verify you get the given y-value. This direct approach eliminates guesswork and builds confidence with function notation.
5. A car's value decreases linearly. It is worth $18{,}000$ when it is 2 years old and $12{,}000$ when it is 5 years old. What is the rate of change in value (dollars per year)?
   Answer: $-2000$ This problem asks for the rate of change in a car's value over time. We have two points: $(2, 18000)$ and $(5, 12000)$, representing (years, value in dollars). The rate of change is the slope: $m = \frac{12000 - 18000}{5 - 2} = \frac{-6000}{3} = -2000$ dollars per year. The negative rate indicates the car is losing value over time, which makes sense for depreciation. This means the car loses $2,000$ in value each year. In word problems involving rates of change, the slope directly represents the rate, and the units come from the y-variable divided by the x-variable units.
6. A line crosses the $y$-axis at $-3$ and also passes through $(4, 1)$. Which equation represents the line?
   Answer: $y=x-3$ This problem provides the y-intercept and one point to find the equation of a line. We know the line crosses the y-axis at -3 and passes through (4, 1). The slope is m = (1 - (-3))/(4 - 0) = 4/4 = 1. Using slope-intercept form with slope 1 and y-intercept -3: y = 1·x + (-3) = x - 3. We can verify with the given point (4, 1): 1 = 4 - 3 = 1 ✓. A common mistake is confusing which point represents the y-intercept; remember that the y-intercept occurs when x = 0. Always double-check your equation by substituting the given point.
7. A linear function $g$ satisfies $g(2)=7$ and $g(6)=19$. What is the value of $g(10)$ ?
   Answer: $31$ When you encounter a linear function problem with two given points, you're finding the equation of a line and using it to determine other values. Linear functions have a constant rate of change, so you can find the slope and then extend the pattern. First, find the slope using the two given points $(2,7)$ and $(6,19)$. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19-7}{6-2} = \frac{12}{4} = 3$. This means the function increases by 3 units for every 1-unit increase in the input. To find $g(10)$, you can extend this pattern from either known point. From $(6,19)$ to $(10,19)$, the input increases by 4, so the output increases by $3 \times 4 = 12$. Therefore, $g(10) = 19 + 12 = 31$. Looking at the wrong answers: Choice B ($25$) might result from incorrectly calculating the slope as 2 instead of 3, giving $19 + 2(4) = 27$, though this doesn't match exactly. Choice C ($28$) could come from using an incorrect slope or arithmetic error. Choice D ($34$) might result from using slope 4 instead of 3, giving $19 + 4(4) = 35$, close but still wrong due to the slope error. The correct answer is A ($31$). Strategy tip: For linear function problems, always calculate the slope carefully using the slope formula, then extend the pattern systematically. Double-check by verifying your slope works with both given points before finding the unknown value.
8. The graph of a linear function $g$ is shown in the $xy$-plane. The function $h$ is defined by $h(x)=g(x-3)+4$. What is the $x$-intercept of the graph of $h$?
   Answer: $(5,0)$ From the graph, line $g$ passes through $(0,-8)$ and $(4,0)$, so it has slope $\frac{0-(-8)}{4-0}=2$ and equation $g(x)=2x-8$. The transformed function is $h(x)=g(x-3)+4=2(x-3)-8+4=2x-6-8+4=2x-10$. To find the $x$-intercept, set $h(x)=0$: $2x-10=0$, so $x=5$. The $x$-intercept is $(5,0)$.
9. A linear function is given by $f(x)=7-2x$. For what value of $x$ does $f(x)=-5$?
   Answer: $6$ This problem asks us to solve for x when a linear function equals a specific value. Given f(x) = 7 - 2x and f(x) = -5, we set up the equation: -5 = 7 - 2x. Adding 2x to both sides: 2x - 5 = 7. Adding 5 to both sides: 2x = 12. Dividing by 2: x = 6. We can verify by substituting back: f(6) = 7 - 2(6) = 7 - 12 = -5 ✓. This type of problem requires basic algebraic manipulation to isolate the variable. Always verify your solution by substituting it back into the original function.
10. A line passes through the points $(2, -1)$ and $(-4, 11)$. What is the equation of the line in slope-intercept form $y = mx + b$?
    Answer: $y = -2x + 3$ We need to find the equation of the line passing through (2, -1) and (-4, 11) in the form y = mx + b. First, calculate the slope: m = (11 - (-1))/(-4 - 2) = 12/(-6) = -2. Now use point-slope form with (2, -1): y - (-1) = -2(x - 2), which gives y + 1 = -2x + 4, so y = -2x + 3. The most common error is sign mistakes when subtracting negative numbers in the slope calculation. Always double-check by substituting both original points back into your final equation to verify they satisfy it.
11. The graph shown displays the height $H$, in inches, of a plant $d$ days after it was planted. The linear model shown has equation $H=0.4d+6$. According to the model, how many days after planting will the plant be exactly $3$ times its original height?
    Answer: $30$ The original height is at $d=0$: $H=0.4(0)+6=6$ inches. Three times this is 18 inches. Setting $0.4d+6=18\Rightarrow 0.4d=12\Rightarrow d=30$ days. Choice B: used 3(6)+6=24, then 0.4d=18. Choice C: found when height doubles. Choice D: used 4×original height.
12. The graph of the linear function $h$ is shown in the $xy$-plane. If $h(x)=mx+b$, what is the value of $h(b)-h(m)$?
    Answer: $-14$ The graph shows slope $m=-2$ and $y$-intercept $b=5$, so $h(x)=-2x+5$. Then $h(b)=h(5)=-2(5)+5=-5$ and $h(m)=h(-2)=-2(-2)+5=9$. Therefore $h(b)-h(m)=-5-9=-14$. Choice B is the magnitude with wrong sign. Choice C omits the constant in one evaluation. Choice D comes from computing $h(b-m)$ instead.
13. What are the coordinates of the point of intersection of the lines $y=2x-3$ and $y=-x+9$ ?
    Answer: $(4,5)$ When you encounter two linear equations and need to find their intersection point, you're looking for the coordinates where both lines meet—the values of $x$ and $y$ that satisfy both equations simultaneously. To solve this system, set the equations equal since both equal $y$: $2x - 3 = -x + 9$. Adding $x$ to both sides gives $3x - 3 = 9$, then adding 3 to both sides yields $3x = 12$, so $x = 4$. Substitute this back into either original equation: $y = 2(4) - 3 = 8 - 3 = 5$. The intersection point is $(4, 5)$. Looking at the wrong answers: Choice B gives $(5, 4)$—this reverses the $x$ and $y$ coordinates, a common error when students mix up their substitution or final answer format. Choice C suggests $(3, 3)$, which you might get if you made an arithmetic error when solving $3x = 12$ or if you incorrectly assumed the intersection has equal coordinates. Choice D gives $(6, 6)$, another "equal coordinates" trap that could result from calculation mistakes or misreading the problem. You can always verify your answer by checking that both coordinates satisfy both original equations: $5 = 2(4) - 3 = 5$ ✓ and $5 = -(4) + 9 = 5$ ✓. Study tip: Always substitute your final answer back into both original equations to confirm it works. This catches calculation errors and builds confidence in your solution method.
14. A line has equation $y=-\tfrac{4}{5}x+6$. What is the $x$-intercept of the line?
    Answer: $\tfrac{15}{2}$ This problem asks for the x-intercept of a given line. The x-intercept occurs where the line crosses the x-axis, which is when y = 0. For the equation y = (-4/5)x + 6, we set y = 0: 0 = (-4/5)x + 6. Subtracting 6 from both sides: -6 = (-4/5)x. Multiplying both sides by -5/4: x = (-6)(-5/4) = 30/4 = 15/2 = 7.5. The x-intercept is 15/2. We can verify: y = (-4/5)(15/2) + 6 = -6 + 6 = 0 ✓. To find x-intercepts, set y = 0 and solve for x; to find y-intercepts, set x = 0 and solve for y.
15. A music streaming service charges a one-time sign-up fee of $6 and then $3 per month. Let $m$ be the number of months after sign-up and $C$ be the total cost in dollars. Which equation represents the relationship between $C$ and $m$? Be careful to distinguish the initial fee from the monthly rate.
    Answer: $C=3m+6$ This question asks for an equation representing the total cost C in terms of months m for a music streaming service. The service has a one-time sign-up fee of $6 (this is the starting cost when m = 0) and charges $3 per month thereafter. The total cost equals the initial fee plus the monthly charge times the number of months: C = 6 + 3m, which can be written as C = 3m + 6. A common error is confusing which value is the coefficient (monthly rate) and which is the constant (initial fee), leading to equations like C = 6m + 3. When dealing with linear cost models, identify the fixed cost (y-intercept) and variable cost (slope) before writing the equation.
16. A line passes through $(2, -1)$ and $(2, 5)$. Which of the following best describes this relation?
    Answer: It is not a function of $x$ This problem asks us to analyze a line passing through two points with the same x-coordinate. The points (2, -1) and (2, 5) both have x = 2, which means this is a vertical line. Vertical lines have undefined slope and cannot be written as functions of x because they fail the vertical line test. For any given x-value (in this case x = 2), there are multiple y-values, which violates the definition of a function. Therefore, this relation is not a function of x. Vertical lines are always written in the form x = constant, not y = mx + b.
17. A taxi company charges according to the formula $f(m)=2.5m+3,$ where $m$ is the number of miles traveled and $f(m)$ is the fare in dollars. How many miles does a passenger travel if the fare is $28 ?
    Answer: $10\text{ miles}$ This problem tests your ability to work with linear functions and solve equations—a core algebra skill. When you're given a function formula and asked to find an input value that produces a specific output, you need to set up and solve an equation. You're told the fare is $28, so you need to solve $f(m) = 28$. Substituting the given formula: $2.5m + 3 = 28$. To solve for $m$, first subtract 3 from both sides: $2.5m = 25$. Then divide both sides by 2.5: $m = \frac{25}{2.5} = 10$. So the passenger traveled 10 miles. Let's check why each answer choice works or doesn't. Choice A (10 miles) is correct because when you substitute $m = 10$ back into the original formula: $f(10) = 2.5(10) + 3 = 25 + 3 = 28$ ✓ Choice B (12.5 miles) gives $f(12.5) = 2.5(12.5) + 3 = 31.25 + 3 = 34.25$, which is too high. Choice C (9 miles) gives $f(9) = 2.5(9) + 3 = 22.5 + 3 = 25.5$, which is too low. Choice D (11 miles) gives $f(11) = 2.5(11) + 3 = 27.5 + 3 = 30.5$, also too high. Strategy tip: Always verify your answer by plugging it back into the original equation. This catches arithmetic errors and confirms you solved correctly. Also, remember that "working backwards" from a given output to find the input is just solving an equation—don't overthink it.
18. On the coordinate plane, a line passes through the points $(0, -2)$ and $(4, 6)$. Which equation represents the line?
    Answer: $y=2x-2$ This problem asks for the equation of a line passing through two given points. First, we find the slope using $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-2)}{4 - 0} = \frac{8}{4} = 2$. Since one point is (0, -2), we can see that the y-intercept is -2. Using slope-intercept form $y = mx + b$, we get $y = 2x + (-2) = 2x - 2$. We can verify this by checking that both points satisfy the equation: for (0, -2): $-2 = 2(0) - 2$ ✓, and for (4, 6): $6 = 2(4) - 2 = 6$ ✓. Always verify your equation by substituting both given points back into it.
19. The table shown gives values of a linear function $f$ at selected inputs. If $f(x)=ax+b$, what is the value of $a+b$?
    Answer: $-2.5$ From the table, using points $(0,-2)$ and $(4,-4)$: slope $a=\frac{-4-(-2)}{4-0}=\frac{-2}{4}=-0.5$. The $y$-intercept is $b=-2$. Therefore $a+b=-0.5+(-2)=-2.5$.
20. A line is shown on the coordinate plane and passes through $ (0, 2) $ and $ (3, 5) $. What is the slope of the line?
    Answer: $1$ This problem asks for the slope of a line passing through two given points. Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ with points $ (0, 2) $ and $ (3, 5) $: $m = \frac{5 - 2}{3 - 0} = \frac{3}{3} = 1$. The slope is 1, meaning the line rises 1 unit vertically for every 1 unit horizontally. We can verify this makes sense: starting at $ (0, 2) $, if we move right 3 units to x = 3, we should move up 3 units to y = 2 + 3 = 5 ✓. When the slope equals 1, the line rises at a 45-degree angle. Always simplify fractions to their lowest terms when expressing slope.

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