Graph Linear and Quadratic Functions - Algebra 1

1. What is the slope of a line through points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $?

   $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for rate of change.

2. What is the $x$-intercept of the line $y=mx+b$ when $m\neq 0$?

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

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

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

4. What does it mean if a line has undefined slope?

   The line is vertical: $x=a$. Infinite slope means no horizontal change as $y$ changes.

5. What is the axis of symmetry of $y=2(x+1)^2-3$?

   $x=-1$. Read $h$-value directly from vertex form.

6. What is the standard form of a linear equation used for intercepts?

   $Ax+By=C$. Form that easily shows intercepts when substituting $x=0$ or $y=0$.

7. What is the vertex $x$-coordinate for $y=2x^2+8x+1$?

   $x=-2$. Use formula $x=-\frac{b}{2a}=-\frac{8}{4}=-2$.

8. What is the slope of the line $3x-6y=12$?

   $\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$ to identify slope.

9. What are the $x$-intercepts of $y=a(x-r_1)(x-r_2)$?

   $(r_1,0)$ and $(r_2,0)$. Where the parabola crosses the $x$-axis.

10. What is the vertex of $y=2(x+1)^2-3$?

    $(-1,-3)$. Read vertex directly from form $y=a(x-h)^2+k$.

11. What is the slope-intercept form of a linear function?

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

12. What is the axis of symmetry for $y=a(x-h)^2+k$?

    $x=h$. Vertical line through the vertex.

13. What are the $x$-intercepts of $y=x^2-9$?

    $(-3,0)$ and $(3,0)$. Factor as $(x-3)(x+3)=0$ to find roots.

14. What is the $y$-intercept of $y=-3(x-2)^2+5$?

    $(0,-7)$. Expand and substitute $x=0$: $y=-3(4)+5=-7$.

15. Does $y=-3(x-2)^2+5$ have a maximum or minimum, and what is its value?

    Maximum value $5$. Since $a=-3<0$, parabola opens down with maximum.

16. What is the $y$-intercept of $y=-\frac{1}{2}(x+4)^2+9$?

    $(0,1)$. Expand and substitute $x=0$: $y=-\frac{1}{2}(16)+9=1$.

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

    $(2,5)$. Read vertex directly from form $y=a(x-h)^2+k$.

18. What does $a>0$ tell you about the parabola $y=ax^2+bx+c$?

    It opens upward; the vertex is a minimum. Positive $a$ creates a U-shaped parabola.

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

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

20. How many $x$-intercepts does $y=(x-1)(x+5)$ have?

    $2$. Two distinct factors give two $x$-intercepts.

21. How many $x$-intercepts does $y=(x+2)^2+1$ have?

    $0$. Vertex form shows parabola shifted up, no $x$-intercepts.

22. Identify the $x$-intercepts of $y=(x-1)(x-4)$.

    $(1,0)$ and $(4,0)$. Set $y=0$ and solve $(x-1)(x-4)=0$.

23. Identify the $y$-intercept of $2x+3y=12$.

    $(0,4)$. Set $x=0$: $2(0)+3y=12$, so $y=4$.

24. What is the standard form of a quadratic function?

    $y=ax^2+bx+c$. General form where $a$ determines opening direction.

25. What is the vertex of $y=x^2-6x+5$?

    $(3,-4)$. Use $x=-\frac{b}{2a}=-\frac{-6}{2}=3$, then find $y$.

26. What is the vertex of $y=a(x-h)^2+k$?

    $(h,k)$. Directly read from vertex form.

27. What is the vertex of $y=x^2+4x+4$?

    $(-2,0)$. Perfect square: $(x+2)^2=0$ has vertex at $x=-2$.

28. What is the minimum or maximum value of $y=2x^2+8x+1$?

    Minimum value $-7$. Since $a=2>0$, parabola has minimum at vertex.

29. What is the vertex of $y=2x^2+8x+1$?

    $(-2,-7)$. Substitute $x=-2$: $y=2(-2)^2+8(-2)+1=-7$.

30. State the vertex directly from $y=-\frac{1}{2}(x+4)^2+9$.

    $(-4,9)$. Read vertex directly from form $y=a(x-h)^2+k$.

31. Identify the $x$-intercept of $2x+3y=12$.

    $(6,0)$. Set $y=0$: $2x+3(0)=12$, so $x=6$.

32. Does $y=-\frac{1}{2}(x+4)^2+9$ have a maximum or minimum, and what is its value?

    Maximum value $9$. Since $a=-\frac{1}{2}<0$, parabola opens down with maximum.

33. What is the $y$-intercept of $y=x^2-6x+5$?

    $(0,5)$. Set $x=0$: $y=0^2-6(0)+5=5$.

34. What is the $x$-intercept of the line $y=-4x+7$?

    $\left(\frac{7}{4},0\right)$. Set $y=0$: $0=-4x+7$, so $x=\frac{7}{4}$.

35. How do you find the $y$-intercept of $y=ax^2+bx+c$?

    Evaluate at $x=0$: $y=c$. Substitute $x=0$ into the function.

36. State the axis of symmetry for $y=-\frac{1}{2}(x+4)^2+9$.

    $x=-4$. Read $h$-value directly from vertex form.

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

    $x=-\frac{b}{2a}$. Formula derived from completing the square.

38. What is the vertex form of a quadratic function?

    $y=a(x-h)^2+k$. Shows vertex $ (h,k) $ and opening direction with $a$.

39. What is the axis of symmetry of $y=x^2-6x+5$?

    $x=3$. Vertical line through the vertex at $x=3$.

40. What is the slope of a line passing through $ (2,5) $ and $ (6,1) $?

    $m=-1$. Use slope formula: $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

41. What is the slope of a line passing through $(2,5)$ and $(6,1)$?

    $m=-1$. Use slope formula: $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

42. What is the minimum or maximum value of $y=a(x-h)^2+k$?

    $k$. The $y$-coordinate of the vertex.

43. What is the vertex of $y=(x-7)(x-1)$?

    $(4,-9)$. Substitute $x=4$ into expanded form $y=x^2-8x+7$.

44. What is the axis of symmetry of $y=(x-7)(x-1)$?

    $x=4$. Average of roots: $\frac{7+1}{2}=4$.

45. What is the equation of the vertical line passing through $x=4$?

    $x=4$. Vertical lines have constant $x$-values.

46. What does $a<0$ tell you about the parabola $y=ax^2+bx+c$?

    It opens downward; the vertex is a maximum. Negative $a$ creates an upside-down parabola.

47. How many $x$-intercepts does $y=(x-3)^2$ have?

    $1$ (a double root at $x=3$). Perfect square touches $x$-axis at one point.

48. Identify the slope and $y$-intercept in $y=\frac{3}{5}x-2$.

    $m=\frac{3}{5}$ and $b=-2$. Directly read from slope-intercept form $y=mx+b$.

49. What is the equation of the horizontal line with $y$-intercept $-3$?

    $y=-3$. Horizontal lines have constant $y$-values.

50. What does it mean if a line has slope $m=0$?

    The line is horizontal: $y=b$. Zero slope means no vertical change as $x$ changes.

51. What is the axis of symmetry of $y=-3(x-2)^2+5$?

    $x=2$. Read $h$-value directly from vertex form.

52. What is the factored form of a quadratic with roots $r_1$ and $r_2$?

    $y=a(x-r_1)(x-r_2)$. Shows $x$-intercepts directly as $r_1$ and $r_2$.

53. What is the $y$-intercept of the line $y=-4x+7$?

    $(0,7)$. The constant term when $x=0$.

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