# SAT Linear Functions Worksheet PDF with Answers | 1600.now Source: https://1600.now/sat-linear-functions-worksheet --- ## SAT Linear Functions Worksheet A linear function changes by a constant rate. In $f(x)=mx+b$, $m$ is the output change per input unit and $b=f(0)$ is the starting output. Two values determine that rate even when neither input is zero. Written by [Luke Finigan](https://1600.now/about) The 12 questions below include slopes, intercepts, tables, and linear word problems. Choose a student PDF or the Letter/A4 version with explained answers. 12 questions · about 25 minutes · Algebra › Linear functions [Download 4-page PDF](https://1600.now/downloads/worksheets/sat-linear-functions-worksheet-student-letter.pdf) [Share to Google Classroom](https://classroom.google.com/share?url=https%3A%2F%2F1600.now%2Fsat-linear-functions-worksheet) · This worksheet is free to print and share under [CC BY-NC-ND 4.0](https://creativecommons.org/licenses/by-nc-nd/4.0/). Tutors may also use it in paid sessions. If you post it online, please link to this page instead of re-uploading the PDF. These are original questions, not College Board items. ### Move from a known value using the rate Given $(x_1,y_1)$ and $(x_2,y_2)$, calculate $m=\dfrac{y_2-y_1}{x_2-x_1}$. Keep the subtraction order the same in numerator and denominator. Read the slope in context: liters per minute, dollars per visit, or another output unit per input unit. You can find a new output directly from a known point: $f(v)=f(u)+m(v-u)$. Going left makes the input change negative. With a negative slope, moving left increases the output. If you need the equation, substitute a known point into $y=mx+b$ to solve for $b$. A value in the table at $x=1$ is not the intercept. For an $x$-intercept, solve $f(x)=0$; for a $y$-intercept, evaluate $f(0)$. Try it yourself: A linear function has $f(2)=7$ and $f(5)=16$. What is $f(0)$? · A $1$ B $3$ C $7$ ### Questions 1. **Question 1** What is the slope of $y = 4x - 7$? A. −7 B. −4 C. 4 D. 7 **C.** In slope-intercept form $y = mx + b$, the coefficient of $x$ is the slope. Here the coefficient is 4. 2. **Question 2** A line passes through (2, 5) and (6, 17). What is its slope? A. 2 B. 3 C. 4 D. 12 **B.** Slope is change in $y$ divided by change in $x$: $\dfrac{17 - 5}{6 - 2} = \dfrac{12}{4} = 3$. 3. **Question 3** Which equation describes a line with slope −2 and $y$-intercept 6? A. $y=6x-2$ B. $y=-2x+6$ C. $y=2x+6$ D. $y=-6x+2$ **B.** Substitute $m=-2$ and $b=6$ into $y=mx+b$. The coefficient controls slope and the constant controls the $y$-intercept. 4. **Question 4** A taxi fare, in dollars, is modeled by $\mathrm{C}(m)=3m+5$, where $m$ is the number of miles traveled. What does 5 represent in this model? A. Cost per mile B. Miles per dollar C. Initial charge in dollars D. Maximum trip length **C.** At $m=0$, $\mathrm{C}(0)=5$. The constant is the starting charge before the distance-based charge is added. 5. **Question 5** If $f$ is linear, $f(1)=7$, and $f(4)=16$, what is $f(6)$? A. 18 B. 20 C. 22 D. 24 **C.** The slope is $\dfrac{16-7}{4-1}=3$. Moving from $x=4$ to $x=6$ adds $2(3)=6$, giving 22. 6. **Question 6** What is the $x$-intercept of $y=3x-12$? A. (−12, 0) B. (0, −12) C. (3, 0) D. (4, 0) **D.** At an $x$-intercept, $y=0$. Solving $0=3x-12$ gives $x=4$, so the point is (4, 0). 7. **Question 7** A tank contains 90 liters at time $t=0$ and 66 liters at $t=4$ minutes. If the rate is constant, which model gives the remaining liters? A. $\mathrm{V}(t)=90-6t$ B. $\mathrm{V}(t)=90-24t$ C. $\mathrm{V}(t)=66-6t$ D. $\mathrm{V}(t)=6t+90$ **A.** The volume drops 24 liters in 4 minutes, a rate of −6 liters per minute. With initial volume 90, $\mathrm{V}(t)=90-6t$. 8. **Question 8** Which line is parallel to $y=5x+2$ and passes through (1, 9)? A. $y=-5x+14$ B. $y=5x+4$ C. $y=2x+7$ D. $y=5x+9$ **B.** A parallel line has slope 5. Substituting (1, 9) gives $9=5+b$, so $b=4$. 9. **Question 9** | $x$ | $f(x)$ | | --- | --- | | 2 | 8 | | 4 | 14 | | 6 | 20 | The table shows values of a linear function. What is $f(0)$? A. −2 B. 0 C. 2 D. 4 **C.** The output increases by 6 when $x$ increases by 2, so the slope is 3. From $f(2)=8$, subtract 2(3) to get $f(0)=2$. 10. **Question 10** For $f(x)=-0.5x+12$, how does $f(x)$ change when $x$ increases by 8? A. Decreases by 4 B. Decreases by 8 C. Increases by 4 D. Increases by 8 **A.** Multiply the input change by the slope: $-0.5(8)=-4$. The output decreases by 4. 11. **Question 11** A membership costs $24 plus $6 per visit. What is the greatest whole number of visits possible with $90? A. 10 B. 11 C. 12 D. 15 **B.** After the membership fee, $66 remains. At $6 per visit, $\dfrac{66}{6}=11$ visits fit the budget. 12. **Question 12** A linear function $g$ satisfies $g(3)=2$ and $g(7)=-6$. What is $g(1)$? A. −2 B. 0 C. 4 D. 6 **D.** The slope is $\dfrac{-6-2}{7-3}=-2$. Moving two units left from $x=3$ increases the output by 4, giving $g(1)=6$. 0/12 correct ### Check your linear functions answers Each row is a wrong answer choice from one of the sheet's problems. Find the one you picked, then run the check in the last column. | What your answer looked like | What actually happened | Fix it next time | | --- | --- | --- | | 12 or 4 in problem 2 | 12 is the change in $y$ and 4 is the change in $x$. The slope is one divided by the other. | $\dfrac{17 - 5}{6 - 2} = \dfrac{12}{4} = 3$. | | 18 in problem 5 | You added the change in $x$, 2, to $f(4) = 16$. Each 1-unit step in $x$ adds the slope, 3. | Two steps from $x = 4$ to $x = 6$ add $2 \times 3 = 6$, so $f(6) = 22$. Check with $f(x) = 3x + 4$: $f(1) = 7$ and $f(4) = 16$. | | $(0, -12)$ for the x-intercept of $y = 3x - 12$ (problem 6) | That's the y-intercept, where $x = 0$. | An x-intercept has $y = 0$: $0 = 3x - 12$, so $x = 4$ and the point is $(4, 0)$. | | $V(t) = 90 - 24t$ in problem 7 | 24 liters is the drop over all 4 minutes. The coefficient of $t$ has to be the drop per minute. | $\dfrac{24}{4} = 6$ liters per minute, so $V(t) = 90 - 6t$. Check: $V(4) = 90 - 24 = 66$, while the other model gives $V(4) = -6$. | | $y = 5x + 9$ in problem 8 | 9 is the $y$-value at $x = 1$, not at $x = 0$, so it isn't the y-intercept. | Keep the slope 5 and solve for the intercept: $9 = 5(1) + b$, so $b = 4$ and the line is $y = 5x + 4$. | | 15 in problem 11 | $\dfrac{90}{6} = 15$ spends the whole budget on visits and skips the $24 membership fee. | Take the fee out first: $90 - 24 = 66$, and $\dfrac{66}{6} = 11$ visits. Check: $24 + 6(11) = 90$. | #### Worked example: moving along a line from two values Problem 12 gives $g(3) = 2$ and $g(7) = -6$ and asks for $g(1)$. Find the slope first: $\dfrac{-6 - 2}{7 - 3} = \dfrac{-8}{4} = -2$. The output drops 2 for every step to the right. Going from $x = 3$ to $x = 1$ is two steps to the left, so the output rises by $2 \times 2 = 4$: $g(1) = 2 + 4 = 6$. Check with the equation $g(x) = -2x + 8$: $g(3) = 2$, $g(7) = -6$, and $g(1) = 6$. The choice $-2$ is the slope, and it's also what you get by moving the wrong way. The choice 4 is the change in output, and 0 is $g(4)$. ### Apply the check to fresh questions For a fresh linear-function question, attach units to the slope and check a second known point. Then use the practice link for more tables and models. 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