Use Factoring, Squares to Analyze Graphs - Algebra
Card 1 of 30
Factor $f(x)=x^2-9$ to show its zeros.
Factor $f(x)=x^2-9$ to show its zeros.
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$f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.
$f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.
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Identify the axis of symmetry for $f(x)=(x-7)(x-1)$.
Identify the axis of symmetry for $f(x)=(x-7)(x-1)$.
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$x=4$. Midpoint of zeros: $\frac{7+1}{2}=4$.
$x=4$. Midpoint of zeros: $\frac{7+1}{2}=4$.
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What number do you add and subtract to complete the square for $x^2+bx$?
What number do you add and subtract to complete the square for $x^2+bx$?
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Add and subtract $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, squared.
Add and subtract $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, squared.
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Identify the zeros of $f(x)=2x(x-5)$.
Identify the zeros of $f(x)=2x(x-5)$.
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$x=0$ and $x=5$. Set each factor equal to zero and solve.
$x=0$ and $x=5$. Set each factor equal to zero and solve.
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What is the extreme value of $f(x)=a(x-h)^2+k$ in terms of $k$ and $a$?
What is the extreme value of $f(x)=a(x-h)^2+k$ in terms of $k$ and $a$?
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Extreme value is $k$ (min if $a>0$, max if $a<0$). Value $k$ is minimum when $a>0$, maximum when $a<0$.
Extreme value is $k$ (min if $a>0$, max if $a<0$). Value $k$ is minimum when $a>0$, maximum when $a<0$.
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What does the sign of $a$ tell you about the opening of $f(x)=ax^2+bx+c$?
What does the sign of $a$ tell you about the opening of $f(x)=ax^2+bx+c$?
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$a>0$ opens up; $a<0$ opens down. Coefficient $a$ determines parabola direction.
$a>0$ opens up; $a<0$ opens down. Coefficient $a$ determines parabola direction.
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What is the factored (intercept) form of a quadratic function with zeros $r_1$ and $r_2$?
What is the factored (intercept) form of a quadratic function with zeros $r_1$ and $r_2$?
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$f(x)=a(x-r_1)(x-r_2)$. Factored form directly shows zeros at $x=r_1$ and $x=r_2$.
$f(x)=a(x-r_1)(x-r_2)$. Factored form directly shows zeros at $x=r_1$ and $x=r_2$.
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What is the axis of symmetry for $f(x)=a(x-h)^2+k$?
What is the axis of symmetry for $f(x)=a(x-h)^2+k$?
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$x=h$. Vertical line through vertex at $x=h$.
$x=h$. Vertical line through vertex at $x=h$.
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What is the axis of symmetry for $f(x)=ax^2+bx+c$?
What is the axis of symmetry for $f(x)=ax^2+bx+c$?
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$x=-\frac{b}{2a}$. Formula derived from vertex $x$-coordinate.
$x=-\frac{b}{2a}$. Formula derived from vertex $x$-coordinate.
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What does $\Delta=b^2-4ac=0$ imply about the zeros of $ax^2+bx+c$?
What does $\Delta=b^2-4ac=0$ imply about the zeros of $ax^2+bx+c$?
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One real double zero. Zero discriminant gives one $x$-intercept (vertex on axis).
One real double zero. Zero discriminant gives one $x$-intercept (vertex on axis).
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What does $\Delta=b^2-4ac>0$ imply about the zeros of $ax^2+bx+c$?
What does $\Delta=b^2-4ac>0$ imply about the zeros of $ax^2+bx+c$?
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Two distinct real zeros. Positive discriminant gives two $x$-intercepts.
Two distinct real zeros. Positive discriminant gives two $x$-intercepts.
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What is the discriminant used to determine the number of real zeros of $ax^2+bx+c$?
What is the discriminant used to determine the number of real zeros of $ax^2+bx+c$?
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$\Delta=b^2-4ac$. Formula under square root in quadratic formula.
$\Delta=b^2-4ac$. Formula under square root in quadratic formula.
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What is the symmetry relationship between zeros $r_1$ and $r_2$ and the axis of symmetry?
What is the symmetry relationship between zeros $r_1$ and $r_2$ and the axis of symmetry?
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Axis is midpoint: $x=\frac{r_1+r_2}{2}$. Axis is equidistant from both zeros.
Axis is midpoint: $x=\frac{r_1+r_2}{2}$. Axis is equidistant from both zeros.
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What are the zeros of $f(x)=a(x-r_1)(x-r_2)$?
What are the zeros of $f(x)=a(x-r_1)(x-r_2)$?
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$x=r_1$ and $x=r_2$. Values that make each factor equal zero.
$x=r_1$ and $x=r_2$. Values that make each factor equal zero.
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What is the $y$-intercept of $f(x)=ax^2+bx+c$?
What is the $y$-intercept of $f(x)=ax^2+bx+c$?
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$(0,c)$. Where parabola crosses $y$-axis at $x=0$.
$(0,c)$. Where parabola crosses $y$-axis at $x=0$.
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Factor $f(x)=x^2+7x+12$ to show its zeros.
Factor $f(x)=x^2+7x+12$ to show its zeros.
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$f(x)=(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$.
$f(x)=(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$.
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Factor $f(x)=x^2-9$ to show its zeros.
Factor $f(x)=x^2-9$ to show its zeros.
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$f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.
$f(x)=(x-3)(x+3)$. Difference of squares: $a^2-b^2=(a-b)(a+b)$.
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Find the maximum value of $f(x)=-2x^2+8x+1$.
Find the maximum value of $f(x)=-2x^2+8x+1$.
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Maximum value is $9$. Since $a=-2<0$, vertex gives maximum.
Maximum value is $9$. Since $a=-2<0$, vertex gives maximum.
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Find the minimum value of $f(x)=x^2-10x+30$.
Find the minimum value of $f(x)=x^2-10x+30$.
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Minimum value is $5$. Complete square to find vertex $y$-coordinate.
Minimum value is $5$. Complete square to find vertex $y$-coordinate.
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Find the vertex of $f(x)=2x^2+12x+1$ without graphing.
Find the vertex of $f(x)=2x^2+12x+1$ without graphing.
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$(-3,-17)$. Axis at $x=-3$; substitute to get $y=-17$.
$(-3,-17)$. Axis at $x=-3$; substitute to get $y=-17$.
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Find the vertex of $f(x)=x^2-4x+9$ without graphing.
Find the vertex of $f(x)=x^2-4x+9$ without graphing.
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$(2,5)$. Axis at $x=2$; substitute to get $y=5$.
$(2,5)$. Axis at $x=2$; substitute to get $y=5$.
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Rewrite in vertex form by completing the square: $f(x)=-x^2+6x-2$.
Rewrite in vertex form by completing the square: $f(x)=-x^2+6x-2$.
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$f(x)=-(x-3)^2+7$. Factor out $-1$, then complete square inside.
$f(x)=-(x-3)^2+7$. Factor out $-1$, then complete square inside.
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Rewrite in vertex form by completing the square: $f(x)=2x^2+8x+3$.
Rewrite in vertex form by completing the square: $f(x)=2x^2+8x+3$.
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$f(x)=2(x+2)^2-5$. Factor out $2$, then complete square inside.
$f(x)=2(x+2)^2-5$. Factor out $2$, then complete square inside.
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Rewrite in vertex form by completing the square: $f(x)=x^2-6x+10$.
Rewrite in vertex form by completing the square: $f(x)=x^2-6x+10$.
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$f(x)=(x-3)^2+1$. Complete square: $x^2-6x=(x-3)^2-9$.
$f(x)=(x-3)^2+1$. Complete square: $x^2-6x=(x-3)^2-9$.
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A model is $f(x)=a(x-r_1)(x-r_2)$ with $a>0$. What does $f(x)>0$ mean between the zeros?
A model is $f(x)=a(x-r_1)(x-r_2)$ with $a>0$. What does $f(x)>0$ mean between the zeros?
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It is false; between zeros $f(x)<0$ when $a>0$. When $a>0$, parabola is below $x$-axis between zeros.
It is false; between zeros $f(x)<0$ when $a>0$. When $a>0$, parabola is below $x$-axis between zeros.
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Rewrite in vertex form by completing the square: $f(x)=x^2+4x-1$.
Rewrite in vertex form by completing the square: $f(x)=x^2+4x-1$.
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$f(x)=(x+2)^2-5$. Complete square: $x^2+4x=(x+2)^2-4$.
$f(x)=(x+2)^2-5$. Complete square: $x^2+4x=(x+2)^2-4$.
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Complete the square: what is $x^2-12x+20$ in vertex form?
Complete the square: what is $x^2-12x+20$ in vertex form?
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$(x-6)^2-16$. Complete square: $(x-6)^2-36+20=(x-6)^2-16$.
$(x-6)^2-16$. Complete square: $(x-6)^2-36+20=(x-6)^2-16$.
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Identify the zeros of $f(x)=(x-2)(x+5)$.
Identify the zeros of $f(x)=(x-2)(x+5)$.
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$x=2$ and $x=-5$. Set each factor equal to zero.
$x=2$ and $x=-5$. Set each factor equal to zero.
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Complete the square: what is $x^2+2x+7$ in vertex form?
Complete the square: what is $x^2+2x+7$ in vertex form?
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$(x+1)^2+6$. Complete square on $x^2+2x$, then add $6$.
$(x+1)^2+6$. Complete square on $x^2+2x$, then add $6$.
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Complete the square: what is $x^2-8x$ rewritten as a square minus a constant?
Complete the square: what is $x^2-8x$ rewritten as a square minus a constant?
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$(x-4)^2-16$. Add and subtract $(\frac{-8}{2})^2=16$.
$(x-4)^2-16$. Add and subtract $(\frac{-8}{2})^2=16$.
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