Graph Linear and Quadratic Functions - Algebra
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What is the axis of symmetry for $y=a(x-h)^2+k$?
What is the axis of symmetry for $y=a(x-h)^2+k$?
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$x=h$. Vertical line through the vertex.
$x=h$. Vertical line through the vertex.
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What is the vertex of $y=x^2+4x+4$?
What is the vertex of $y=x^2+4x+4$?
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$(-2,0)$. Perfect square: $(x+2)^2=0$ has vertex at $x=-2$.
$(-2,0)$. Perfect square: $(x+2)^2=0$ has vertex at $x=-2$.
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What is the axis of symmetry of $y=(x-7)(x-1)$?
What is the axis of symmetry of $y=(x-7)(x-1)$?
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$x=4$. Average of roots: $\frac{7+1}{2}=4$.
$x=4$. Average of roots: $\frac{7+1}{2}=4$.
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What are the $x$-intercepts of $y=a(x-r_1)(x-r_2)$?
What are the $x$-intercepts of $y=a(x-r_1)(x-r_2)$?
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$(r_1,0)$ and $(r_2,0)$. Where the parabola crosses the $x$-axis.
$(r_1,0)$ and $(r_2,0)$. Where the parabola crosses the $x$-axis.
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What is the vertex form of a quadratic function?
What is the vertex form of a quadratic function?
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$y=a(x-h)^2+k$. Shows vertex $ (h,k) $ and opening direction with $a$.
$y=a(x-h)^2+k$. Shows vertex $ (h,k) $ and opening direction with $a$.
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What is the standard form of a quadratic function?
What is the standard form of a quadratic function?
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$y=ax^2+bx+c$. General form where $a$ determines opening direction.
$y=ax^2+bx+c$. General form where $a$ determines opening direction.
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What is the factored form of a quadratic with roots $r_1$ and $r_2$?
What is the factored form of a quadratic with roots $r_1$ and $r_2$?
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$y=a(x-r_1)(x-r_2)$. Shows $x$-intercepts directly as $r_1$ and $r_2$.
$y=a(x-r_1)(x-r_2)$. Shows $x$-intercepts directly as $r_1$ and $r_2$.
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What is the axis of symmetry for $y=ax^2+bx+c$?
What is the axis of symmetry for $y=ax^2+bx+c$?
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$x=-\frac{b}{2a}$. Formula derived from completing the square.
$x=-\frac{b}{2a}$. Formula derived from completing the square.
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What is the vertex of $y=2x^2+8x+1$?
What is the vertex of $y=2x^2+8x+1$?
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$(-2,-7)$. Substitute $x=-2$: $y=2(-2)^2+8(-2)+1=-7$.
$(-2,-7)$. Substitute $x=-2$: $y=2(-2)^2+8(-2)+1=-7$.
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What is the vertex of $y=a(x-h)^2+k$?
What is the vertex of $y=a(x-h)^2+k$?
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$(h,k)$. Directly read from vertex form.
$(h,k)$. Directly read from vertex form.
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How do you find the $y$-intercept of $y=ax^2+bx+c$?
How do you find the $y$-intercept of $y=ax^2+bx+c$?
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Evaluate at $x=0$: $y=c$. Substitute $x=0$ into the function.
Evaluate at $x=0$: $y=c$. Substitute $x=0$ into the function.
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What is the slope of a line through points $ (x_1, y_1) $ and $ (x_2, y_2) $?
What is the slope of a line through points $ (x_1, y_1) $ and $ (x_2, y_2) $?
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$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for rate of change.
$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for rate of change.
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What is the slope of a line passing through $ (2,5) $ and $ (6,1) $?
What is the slope of a line passing through $ (2,5) $ and $ (6,1) $?
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$m=-1$. Use slope formula: $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.
$m=-1$. Use slope formula: $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.
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What does $a>0$ tell you about the parabola $y=ax^2+bx+c$?
What does $a>0$ tell you about the parabola $y=ax^2+bx+c$?
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It opens upward; the vertex is a minimum. Positive $a$ creates a U-shaped parabola.
It opens upward; the vertex is a minimum. Positive $a$ creates a U-shaped parabola.
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What does $a<0$ tell you about the parabola $y=ax^2+bx+c$?
What does $a<0$ tell you about the parabola $y=ax^2+bx+c$?
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It opens downward; the vertex is a maximum. Negative $a$ creates an upside-down parabola.
It opens downward; the vertex is a maximum. Negative $a$ creates an upside-down parabola.
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What is the minimum or maximum value of $y=a(x-h)^2+k$?
What is the minimum or maximum value of $y=a(x-h)^2+k$?
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$k$. The $y$-coordinate of the vertex.
$k$. The $y$-coordinate of the vertex.
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What is the vertex of $y=x^2-6x+5$?
What is the vertex of $y=x^2-6x+5$?
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$(3,-4)$. Use $x=-\frac{b}{2a}=-\frac{-6}{2}=3$, then find $y$.
$(3,-4)$. Use $x=-\frac{b}{2a}=-\frac{-6}{2}=3$, then find $y$.
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What is the axis of symmetry of $y=x^2-6x+5$?
What is the axis of symmetry of $y=x^2-6x+5$?
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$x=3$. Vertical line through the vertex at $x=3$.
$x=3$. Vertical line through the vertex at $x=3$.
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What is the $y$-intercept of $y=x^2-6x+5$?
What is the $y$-intercept of $y=x^2-6x+5$?
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$(0,5)$. Set $x=0$: $y=0^2-6(0)+5=5$.
$(0,5)$. Set $x=0$: $y=0^2-6(0)+5=5$.
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Identify the $x$-intercepts of $y=(x-1)(x-4)$.
Identify the $x$-intercepts of $y=(x-1)(x-4)$.
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$(1,0)$ and $(4,0)$. Set $y=0$ and solve $(x-1)(x-4)=0$.
$(1,0)$ and $(4,0)$. Set $y=0$ and solve $(x-1)(x-4)=0$.
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What is the vertex of $y=2(x+1)^2-3$?
What is the vertex of $y=2(x+1)^2-3$?
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$(-1,-3)$. Read vertex directly from form $y=a(x-h)^2+k$.
$(-1,-3)$. Read vertex directly from form $y=a(x-h)^2+k$.
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What is the axis of symmetry of $y=2(x+1)^2-3$?
What is the axis of symmetry of $y=2(x+1)^2-3$?
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$x=-1$. Read $h$-value directly from vertex form.
$x=-1$. Read $h$-value directly from vertex form.
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Does $y=-3(x-2)^2+5$ have a maximum or minimum, and what is its value?
Does $y=-3(x-2)^2+5$ have a maximum or minimum, and what is its value?
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Maximum value $5$. Since $a=-3<0$, parabola opens down with maximum.
Maximum value $5$. Since $a=-3<0$, parabola opens down with maximum.
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What is the vertex of $y=-3(x-2)^2+5$?
What is the vertex of $y=-3(x-2)^2+5$?
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$(2,5)$. Read vertex directly from form $y=a(x-h)^2+k$.
$(2,5)$. Read vertex directly from form $y=a(x-h)^2+k$.
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What is the axis of symmetry of $y=-3(x-2)^2+5$?
What is the axis of symmetry of $y=-3(x-2)^2+5$?
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$x=2$. Read $h$-value directly from vertex form.
$x=2$. Read $h$-value directly from vertex form.
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What is the $y$-intercept of $y=-3(x-2)^2+5$?
What is the $y$-intercept of $y=-3(x-2)^2+5$?
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$(0,-7)$. Expand and substitute $x=0$: $y=-3(4)+5=-7$.
$(0,-7)$. Expand and substitute $x=0$: $y=-3(4)+5=-7$.
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What are the $x$-intercepts of $y=x^2-9$?
What are the $x$-intercepts of $y=x^2-9$?
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$(-3,0)$ and $(3,0)$. Factor as $(x-3)(x+3)=0$ to find roots.
$(-3,0)$ and $(3,0)$. Factor as $(x-3)(x+3)=0$ to find roots.
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How many $x$-intercepts does $y=(x+2)^2+1$ have?
How many $x$-intercepts does $y=(x+2)^2+1$ have?
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$0$. Vertex form shows parabola shifted up, no $x$-intercepts.
$0$. Vertex form shows parabola shifted up, no $x$-intercepts.
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How many $x$-intercepts does $y=(x-3)^2$ have?
How many $x$-intercepts does $y=(x-3)^2$ have?
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$1$ (a double root at $x=3$). Perfect square touches $x$-axis at one point.
$1$ (a double root at $x=3$). Perfect square touches $x$-axis at one point.
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How many $x$-intercepts does $y=(x-1)(x+5)$ have?
How many $x$-intercepts does $y=(x-1)(x+5)$ have?
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$2$. Two distinct factors give two $x$-intercepts.
$2$. Two distinct factors give two $x$-intercepts.
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