Graphs - 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. What is the equation for a line perpendicular to $y = -\frac{1}{4}x + 2$ at $(0, 0)$?

   $y = 4x$. Negative reciprocal slope through given point.

3. Determine the x-intercept of the line $y = 2x - 8$.

   4. Set $y = 0$ and solve: $0 = 2x - 8$, so $x = 4$.

4. Identify the slope of the line $y - 6 = \frac{1}{3}(x - 3)$.

   $\frac{1}{3}$. Coefficient of $(x - 3)$ in point-slope form.

5. Identify whether the line $x = 5$ is vertical or horizontal.

   Vertical. Constant $x$-value creates vertical line.

6. What is the equation of a circle with center $(h, k)$ and radius $r$?

   $(x - h)^2 + (y - k)^2 = r^2$. Distance from center to any point on circle.

7. Which form of a line's equation uses the formula $y - y_1 = m(x - x_1)$?

   Point-slope form. Uses a known point $(x_1, y_1)$ and slope $m$.

8. Identify the vertex of the parabola $y = (x - 2)^2 - 3$.

   $(2, -3)$. Vertex form shows $(h, k)$ as the turning point.

9. Identify the y-intercept in the equation $y = 3x + 4$.

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

10. What is the equation of a horizontal line through point $(a, b)$?

    $y = b$. All points have the same $y$-coordinate.

11. Identify the x-intercept in the equation $2x - 6 = 0$.

    3. Solve for $x$ when $y = 0$.

12. What is the slope formula for a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$?

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Change in $y$ over change in $x$ between two points.

13. What is the slope of a line with equation $y = 0$?

    0. Horizontal line has zero rise over run.

14. What are the coordinates of the y-intercept in the line $3x + 4y = 8$?

    $(0, 2)$. Set $x = 0$ and solve for $y$.

15. Identify the direction of opening for the parabola $y = -x^2 + 4x - 1$.

    Downward. Negative coefficient of $x^2$ opens downward.

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

    $y = mx + b$. Shows slope $m$ and $y$-intercept $b$ directly.

17. Identify the transformation applied: $y = (x - 1)^2 + 2$.

    Right 1 unit, up 2 units. Vertex form shows horizontal and vertical shifts.

18. State the vertex form of a quadratic function.

    $y = a(x - h)^2 + k$. Shows vertex $(h, k)$ and vertical shift $k$.

19. What is the axis of symmetry for the parabola $y = ax^2 + bx + c$?

    $x = -\frac{b}{2a}$. Vertical line through the vertex of the parabola.

20. What is the midpoint formula for the segment connecting $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $?

    $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. Average of the coordinates of the endpoints.

21. What is the radius of the circle $x^2 + y^2 = 16$?

    4. Square root of the constant term.

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

    $Ax + By = C$. Linear equation with integer coefficients and no fractions.

23. What is the point of intersection for lines $y = 2x + 1$ and $y = -x + 4$?

    $(1, 3)$. Solve the system by setting equations equal.

24. What is the equation of a line in slope-intercept form?

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

25. Identify the slope of the line: $3x - 4y = 12$.

    $\frac{3}{4}$. Rearrange to $y = \frac{3}{4}x - 3$ to identify slope.

26. What is the slope of a line parallel to $y = -4x + 7$?

    -4. Parallel lines have identical slopes.

27. Identify the center of the circle $(x - 3)^2 + (y + 4)^2 = 25$.

    $(3, -4)$. Values of $(h, k)$ from standard form.

28. What are the coordinates of the x-intercept in the line $5x - 2y = 10$?

    $(2, 0)$. Set $y = 0$ and solve for $x$.

29. Find the midpoint of the segment joining $(1, 2)$ and $(5, 6)$.

    $(3, 4)$. Average the x-coordinates and y-coordinates: $\frac{1+5}{2}, \frac{2+6}{2}$.

30. What is the equation of a circle with center $(h, k)$ and radius $r$?

    $(x - h)^2 + (y - k)^2 = r^2$. Standard circle equation with center $(h, k)$ and radius $r$.

31. What is the definition of a function in terms of ordered pairs?

    Each input has exactly one output. Vertical line test ensures unique outputs.

32. Identify the vertex of the parabola $y = (x - 3)^2 + 2$.

    (3, 2). Vertex form $(x - h)^2 + k$ has vertex at $(h, k)$.

33. Identify the radius in the equation $ $(x - 2)^2$ + $(y + 1)^2$ = 49 $.

    $7$. Square root of the constant gives radius.

34. What is the general form of the equation of a circle?

    $x^2 + y^2 + Dx + Ey + F = 0$. Expanded form of circle equation.

35. What is the vertex of the parabola $y = 3(x + 2)^2 - 5$?

    $(-2, -5)$. Vertex form identifies $(h, k)$ directly.

36. Identify the intercepts of the quadratic $y = (x - 3)(x + 2)$.

    x-intercepts: $3, -2$. Set $y = 0$ to find where parabola crosses $x$-axis.

37. State the formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.

    $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Pythagorean theorem applied to coordinate plane.

38. What does the slope of a line represent in a graph?

    The rate of change or steepness of the line. Measures how much $y$ changes per unit increase in $x$.

39. What are the intercepts in the equation $y = \frac{1}{2}x - 3$?

    x-intercept: 6, y-intercept: -3. Set each variable to zero alternately.

40. What is the general form of the equation of a parabola?

    $y = ax^2 + bx + c$. Quadratic with degree 2 polynomial.

41. What is the equation for a line parallel to $y = 3x - 2$ passing through $(1, 4)$?

    $y = 3x + 1$. Same slope, substitute point to find $b$.

42. What is the equation of a vertical line through point $(a, b)$?

    $x = a$. All points have the same $x$-coordinate.

43. Identify the radius in the equation $(x - 2)^2 + (y + 1)^2 = 49$.

    7. Square root of the constant gives radius.

44. Which point lies on the line $y = 2x + 3$: (1, 5) or (2, 4)?

    (1, 5). Substituting: $y = 2(1) + 3 = 5$, so point $(1, 5)$ satisfies the equation.

45. What is the slope of a line perpendicular to $y = 2x + 3$?

    $-\frac{1}{2}$. Negative reciprocal of the original slope.

46. Find the y-intercept of the line: $4x + 5y = 20$.

    4. Set $x = 0$: $5y = 20$, so $y = 4$.

47. Identify the slope of the line given by $2y - 4x = 8$.

    2. Rearrange to $y = mx + b$ form first.

48. What type of graph represents the equation $y = x^2$?

    Parabola. Quadratic function creates U-shaped curve.

49. Identify the slope in the equation $y = -5x + 2$.

    -5. Coefficient of $x$ in slope-intercept form.

50. What is the general form of a quadratic function?

    $ax^2 + bx + c = 0$. Standard form with highest power of 2.

51. What is the formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$?

    $\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Pythagorean theorem applied to coordinate plane differences.

52. What is the formula for converting a line from standard to slope-intercept form?

    Solve $Ax + By = C$ for $y$. Isolate $y$ by dividing by coefficient of $y$.

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