Algebra Flashcards: Graph Linear And Quadratic Functions

What this deck covers

This deck focuses on Graph Linear And Quadratic Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: What is the slope of a line through points $(x_1, y_1)$ and $(x_2, y_2)$?

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

Flashcard 3: What is the $x$-intercept of the line $y=mx+b$ when $m\neq 0$?

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

Flashcard 4: What is the $y$-intercept of the line $y=mx+b$?

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

Flashcard 5: What does it mean if a line has undefined slope?

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

Flashcard 6: What is the vertex of $y=x^2+4x+4$?

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

Flashcard 7: What is the axis of symmetry of $y=2(x+1)^2-3$?

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

Flashcard 8: What is the standard form of a linear equation used for intercepts?

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

Flashcard 9: What is the slope-intercept form of a linear function?

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

Flashcard 10: What is the vertex $x$-coordinate for $y=2x^2+8x+1$?

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

Flashcard 11: State the axis of symmetry for $y=-\frac{1}{2}(x+4)^2+9$.

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

Flashcard 12: What is the slope of the line $3x-6y=12$?

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

Flashcard 13: What are the $x$-intercepts of $y=a(x-r_1)(x-r_2)$?

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

Flashcard 14: What is the vertex of $y=2(x+1)^2-3$?

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

Flashcard 15: What is the standard form of a linear equation used for intercepts?

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

Flashcard 16: What is the slope-intercept form of a linear function?

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

Flashcard 17: What is the axis of symmetry for $y=a(x-h)^2+k$?

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

Flashcard 18: What are the $x$-intercepts of $y=x^2-9$?

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

Flashcard 19: What is the $y$-intercept of $y=-3(x-2)^2+5$?

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

Flashcard 20: Does $y=-3(x-2)^2+5$ have a maximum or minimum, and what is its value?

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

Flashcard 21: What is the $y$-intercept of $y=-\frac{1}{2}(x+4)^2+9$?

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

Flashcard 22: What is the vertex of $y=-3(x-2)^2+5$?

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

Flashcard 23: What is the $y$-intercept of the line $y=mx+b$?

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

Flashcard 24: What does $a>0$ tell you about the parabola $y=ax^2+bx+c$?

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

Flashcard 25: What does it mean if a line has slope $m=0$?

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

Flashcard 26: What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

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

Flashcard 27: How many $x$-intercepts does $y=(x+2)^2+1$ have?

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

Flashcard 28: What is the $y$-intercept of $y=x^2-6x+5$?

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

Flashcard 29: How many $x$-intercepts does $y=(x-1)(x+5)$ have?

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

Flashcard 30: Does $y=-\frac{1}{2}(x+4)^2+9$ have a maximum or minimum, and what is its value?

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

Flashcard 31: How many $x$-intercepts does $y=(x+2)^2+1$ have?

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

Flashcard 32: Identify the $x$-intercepts of $y=(x-1)(x-4)$.

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

Flashcard 33: Identify the $y$-intercept of $2x+3y=12$.

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

Flashcard 34: What is the standard form of a quadratic function?

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

Flashcard 35: How many $x$-intercepts does $y=(x-3)^2$ have?

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

Flashcard 36: What is the vertex of $y=x^2-6x+5$?

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

Flashcard 37: What is the vertex of $y=(x-7)(x-1)$?

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

Flashcard 38: What is the vertex of $y=a(x-h)^2+k$?

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

Flashcard 39: What is the vertex of $y=x^2+4x+4$?

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

Flashcard 40: What is the minimum or maximum value of $y=2x^2+8x+1$?

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

Flashcard 41: What is the vertex of $y=2x^2+8x+1$?

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

Flashcard 42: State the vertex directly from $y=-\frac{1}{2}(x+4)^2+9$.

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

Flashcard 43: Identify the $x$-intercept of $2x+3y=12$.

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

Flashcard 44: Does $y=-\frac{1}{2}(x+4)^2+9$ have a maximum or minimum, and what is its value?

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

Flashcard 45: What is the $y$-intercept of $y=x^2-6x+5$?

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

Flashcard 46: What is the slope of a line through points $(x_1, y_1)$ and $(x_2, y_2)$?

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

Flashcard 47: What is the $x$-intercept of the line $y=-4x+7$?

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

Flashcard 48: What does $a>0$ tell you about the parabola $y=ax^2+bx+c$?

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

Flashcard 49: What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

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

Flashcard 50: What is the $y$-intercept of $y=-\frac{1}{2}(x+4)^2+9$?

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

Flashcard 51: How do you find the $y$-intercept of $y=ax^2+bx+c$?

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

Flashcard 52: State the axis of symmetry for $y=-\frac{1}{2}(x+4)^2+9$.

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

Flashcard 53: What is the axis of symmetry for $y=ax^2+bx+c$?

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

Flashcard 54: What is the vertex form of a quadratic function?

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

Flashcard 55: What are the $x$-intercepts of $y=x^2-9$?

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

Flashcard 56: What is the axis of symmetry of $y=x^2-6x+5$?

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

Flashcard 57: What is the slope of a line passing through $(2,5)$ and $(6,1)$?

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

Flashcard 58: What is the vertex of $y=2x^2+8x+1$?

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

Flashcard 59: What is the slope of a line passing through $(2,5)$ and $(6,1)$?

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

Flashcard 60: What is the vertex form of a quadratic function?

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

Flashcard 61: What is the minimum or maximum value of $y=a(x-h)^2+k$?

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

Flashcard 62: Identify the slope and $y$-intercept in $y=\frac{3}{5}x-2$.

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

Flashcard 63: Identify the $x$-intercept of $2x+3y=12$.

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

Flashcard 64: What is the vertex of $y=(x-7)(x-1)$?

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

Flashcard 65: What is the axis of symmetry of $y=(x-7)(x-1)$?

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

Flashcard 66: What is the equation of the vertical line passing through $x=4$?

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

Flashcard 67: What does it mean if a line has undefined slope?

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

Flashcard 68: What does $a<0$ tell you about the parabola $y=ax^2+bx+c$?

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

Flashcard 69: What is the $x$-intercept of the line $y=mx+b$ when $m\neq 0$?

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

Flashcard 70: How many $x$-intercepts does $y=(x-3)^2$ have?

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

Flashcard 71: Identify the slope and $y$-intercept in $y=\frac{3}{5}x-2$.

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

Flashcard 72: What is the equation of the horizontal line with $y$-intercept $-3$?

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

Flashcard 73: What does it mean if a line has slope $m=0$?

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

Flashcard 74: What is the axis of symmetry of $y=-3(x-2)^2+5$?

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

Flashcard 75: What is the factored form of a quadratic with roots $r_1$ and $r_2$?

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

Flashcard 76: What is the $y$-intercept of the line $y=-4x+7$?

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

Flashcard 77: Identify the $y$-intercept of $2x+3y=12$.

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

Flashcard 78: What is the $y$-intercept of $y=-3(x-2)^2+5$?

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