Linear Functions - PSAT Math

1. What is the relationship between slopes of parallel nonvertical lines?

   They are equal: $m_1 = m_2$. Parallel lines have the same slope.

2. What is the equation of a vertical line through $x = a$?

   $x = a$. Vertical lines have constant x-coordinate.

3. Find the equation of the line perpendicular to $y=\frac{1}{4}x-3$ through $(0,2)$.

   $y=-4x+2$. Perpendicular slope is $-4$; passes through $(0,2)$.

4. Find the slope of the line through $(2,5)$ and $(6,1)$.

   $-1$. $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

5. Evaluate the linear function $f(x)=2x-7$ at $x=5$.

   $f(5)=3$. Substitute: $f(5) = 2(5) - 7 = 10 - 7 = 3$.

6. Identify whether $y=7x+1$ is increasing, decreasing, or constant.

   Increasing. Positive slope ($7>0$) means function increases.

7. Find the slope of the line through $(2,7)$ and $(6,-1)$.

   $m=-2$. Apply slope formula: $m = \frac{-1-7}{6-2} = \frac{-8}{4} = -2$.

8. Convert $4x - 5y = 10$ to slope-intercept form.

   $y = \frac{4}{5}x - 2$. Solve for $y$: $y = \frac{4x-10}{5} = \frac{4}{5}x - 2$.

9. What is the slope of a line parallel to a line with slope $m$?

   $m$. Parallel lines have identical slopes.

10. Find the $y$-intercept of the line $2x+3y=12$.

    $4$. Set $x=0$: $3y=12$, so $y=4$.

11. What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?

    $y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.

12. What is the relationship between slopes of perpendicular nonvertical lines?

    They are negative reciprocals: $m_1m_2 = -1$. Product of perpendicular slopes equals $-1$.

13. Identify the slope and $y$-intercept of $y = -3x + 5$.

    $m=-3$, $b=5$. Read directly from $y = mx + b$ form.

14. What is the slope-intercept form of a linear equation?

    $y=mx+b$. Standard form with slope $m$ and $y$-intercept $b$.

15. State the formula for slope between $(x_1,y_1)$ and $(x_2,y_2)$.

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

16. Find the equation of the line parallel to $y=-2x+7$ through $(3,1)$.

    $y=-2x+7$. Same slope $-2$; use point-slope: $y-1=-2(x-3)$.

17. Find the equation in slope-intercept form of the line with slope $4$ and $y$-intercept $-2$.

    $y = 4x - 2$. Substitute $m$ and $b$ into $y = mx + b$.

18. What do $m$ and $b$ represent in $y=mx+b$?

    $m$ is slope; $b$ is $y$-intercept. Slope determines steepness; $y$-intercept is where line crosses $y$-axis.

19. Find the function value $f(3)$ for $f(x)=-2x+9$.

    $3$. $f(3)=-2(3)+9=-6+9=3$.

20. Find the equation in slope-intercept form through $(0,-4)$ and $(2,0)$.

    $y=2x-4$. Slope is $\frac{0-(-4)}{2-0}=2$; $y$-intercept is $-4$.

21. What is the slope of a vertical line (written as $x=c$)?

    Undefined. Vertical lines have infinite steepness (division by zero).

22. What is the slope of a line perpendicular to $y = \frac{2}{3}x - 1$?

    $-\frac{3}{2}$. Negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$.

23. What is the slope of a horizontal line (written as $y=c$)?

    $0$. Horizontal lines have no rise, so slope is zero.

24. Find the slope of the line $5x-2y=10$.

    $\frac{5}{2}$. Rewrite as $y=\frac{5}{2}x-5$; slope is coefficient of $x$.

25. Identify the $x$-intercept in terms of $m$ and $b$ for $y=mx+b$.

    $x=-\frac{b}{m}$ (if $m\ne 0$). Set $y=0$ and solve for $x$.

26. Find the equation in slope-intercept form of the line with slope $3$ and $y$-intercept $-2$.

    $y=3x-2$. Direct substitution into $y=mx+b$ form.

27. Find the $y$-intercept of $2x + 3y = 12$.

    $(0,4)$. Set $x=0$: $3y=12$, so $y=4$.

28. What is the $x$-intercept of $y = mx + b$ in terms of $m$ and $b$ (with $m \ne 0$)?

    $\left(-\frac{b}{m},0\right)$. Set $y=0$ and solve for $x$.

29. What is the standard form of a linear equation?

    $Ax+By=C$. General form where $A$, $B$, and $C$ are constants.

30. Find the $x$-intercept of $3x - 2y = 6$.

    $(2,0)$. Set $y=0$: $3x=6$, so $x=2$.

31. What is the slope of a line perpendicular to a line with slope $m$?

    $-\frac{1}{m}$ (if $m\ne 0$). Perpendicular slopes multiply to $-1$.

32. What is the $y$-intercept of the line $y = mx + b$?

    $(0,b)$. Where the line crosses the y-axis (when $x=0$).

33. Find the equation of the line with slope $-\frac{1}{2}$ passing through $(4,3)$.

    $y = -\frac{1}{2}x + 5$. Use point-slope form: $y - 3 = -\frac{1}{2}(x - 4)$.

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