Linear Functions - SAT Math

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

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

2. If $y = -x + 2$ is shifted right by 2 units, what is the new equation?

   $y = -(x - 2) + 2$. Horizontal shift affects the $x$-term inside parentheses.

3. If $y = 2x + 5$ is shifted up by 3 units, what is the new equation?

   $y = 2x + 8$. Vertical shift changes only the y-intercept.

4. What is the x-intercept of the line $2x + 5y = 10$?

   5. Set $y = 0$ to get $2x = 10$, so $x = 5$.

5. Find the slope of the line through points $ (1, 2) $ and $ (3, 6) $.

   2. $m = \frac{6-2}{3-1} = \frac{4}{2} = 2$

6. Determine if the lines $y = 2x + 3$ and $y = 2x - 4$ are parallel.

   Yes, they are parallel. Both lines have slope 2, so they're parallel.

7. State the standard form of a linear equation.

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

8. Determine if the lines $y = 3x + 1$ and $y = -\frac{1}{3}x + 2$ are perpendicular.

   Yes, they are perpendicular. Slopes are 3 and $-\frac{1}{3}$; their product is -1.

9. Identify the slope of the line parallel to $y = \frac{1}{2}x - 4$.

   $\frac{1}{2}$. Parallel lines have identical slopes.

10. What is the point-slope form of a linear equation?

    $y - y_1 = m(x - x_1)$. Uses a known point $(x_1, y_1)$ and slope $m$.

11. Identify the slope in the equation $y = 3x + 7$.

    3. The coefficient of $x$ is the slope.

12. Find the slope of the line through points $(1, 2)$ and $(3, 6)$.

    2. $m = \frac{6-2}{3-1} = \frac{4}{2} = 2$

13. Convert $y = -2x + 5$ to standard form.

    $2x + y = 5$. Add $2x$ to both sides to get standard form.

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

    $y - y_1 = m(x - x_1)$. Uses a known point $(x_1, y_1)$ and slope $m$.

15. Convert $y = 4x - 3$ to standard form.

    $4x - y = 3$. Move $y$ to left side and arrange coefficients properly.

16. Identify the slope in the equation $y = -3x + 7$.

    -3. The coefficient of $x$ is the slope.

17. What is the equation of a horizontal line through $(3, 7)$?

    $y = 7$. Horizontal lines have constant $y$-values.

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

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

19. Identify the y-intercept in the equation $3y = 12x + 6$.

    2. Divide by 3: $y = 4x + 2$, so y-intercept is 2.

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

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

21. Write the equation of the line with slope 3 and y-intercept -4.

    $y = 3x - 4$. Direct substitution into $y = mx + b$.

22. Determine the y-intercept of the line $y = -4x + 5$.

    5. The constant term when $x = 0$.

23. What is the x-intercept of the line $y = 2x - 4$?

    2. Set $y = 0$ and solve: $0 = 2x - 4$, so $x = 2$.

24. Find the x-intercept of the line $2x + 3y = 6$.

    3. Set $y = 0$ and solve for $x$.

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

    $Ax + By = C$. General form with integer coefficients $A$, $B$, and $C$.

26. State the point-slope form of a linear equation.

    $y - y_1 = m(x - x_1)$. Uses a known point $(x_1, y_1)$ and slope $m$.

27. Identify the y-intercept in the equation $y = 2x - 7$.

    -7. The constant term in slope-intercept form.

28. What is the midpoint formula for points $(x_1, y_1)$ and $(x_2, y_2)$?

    $\bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg)$. Average of coordinates for both $x$ and $y$ values.

29. Identify the slope in the equation $y = -3x + 5$.

    -3. The coefficient of $x$ in slope-intercept form.

30. What is the equation of a vertical line through $(5, -2)$?

    $x = 5$. Vertical lines have constant $x$-values.

31. Given point $(1, 2)$ and slope 4, find the equation in point-slope form.

    $y - 2 = 4(x - 1)$. Substitute point coordinates and slope into formula.

32. Find the y-intercept of the line $5x + 3y = 15$.

    5. Set $x = 0$ to get $3y = 15$, so $y = 5$.

33. What condition must be true for lines to be perpendicular?

    Product of slopes is -1. Negative reciprocals create 90-degree intersections.

34. Find the midpoint of the segment connecting $(2, 3)$ and $(4, 5)$.

    $(3, 4)$. Use midpoint formula with given coordinates.

35. What does the 'm' represent in the equation $y = mx + b$?

    Slope of the line. The coefficient of $x$ that determines steepness and direction.

36. Find the slope of a line through points $(2, 3)$ and $(4, 7)$.

    2. Use $m = \frac{7-3}{4-2} = \frac{4}{2} = 2$.

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

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

38. Calculate the slope of the line $y = -\frac{2}{3}x + 4$.

    $-\frac{2}{3}$. The coefficient of $x$ in slope-intercept form.

39. State the formula for calculating the slope between two points.

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

40. Which term represents the y-intercept in $y = 5x + 9$?

    9. The constant term is the y-intercept.

41. State the formula for calculating slope given two points.

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

42. State the formula for finding the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$.

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Change in $y$ divided by change in $x$.

43. Convert $3x + 4y = 12$ to slope-intercept form.

    $y = -\frac{3}{4}x + 3$. Isolate $y$ by dividing both sides by 4.

44. Convert the equation $3x - 2y = 6$ to slope-intercept form.

    $y = \frac{3}{2}x - 3$. Solve for $y$ by isolating it on one side.

45. What is the slope of any vertical line?

    Undefined. Division by zero makes slope undefined.

46. What does the 'b' represent in the equation $y = mx + b$?

    Y-intercept of the line. The constant term where the line crosses the y-axis.

47. What is the slope of any horizontal line?

    0. No vertical change means zero slope.

48. Determine the slope of the line $3x - 2y = 6$.

    $\frac{3}{2}$. Rearrange to get $y = \frac{3}{2}x - 3$.

49. Find the x-intercept of the line $4x - 8y = 16$.

    4. Set $y = 0$ to get $4x = 16$, so $x = 4$.

50. What condition must be true for lines to be parallel?

    The slopes are equal. Lines with identical slopes never intersect.

51. Find the equation of a line with slope 2 passing through (1, -3).

    $y + 3 = 2(x - 1)$. Substitute the point and slope into point-slope form.

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