Complete the Square to Find Extrema - Algebra
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At what $x$ does $f(x)=x^2+6x+5$ attain its minimum value?
At what $x$ does $f(x)=x^2+6x+5$ attain its minimum value?
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$x=-3$. The vertex $x$-coordinate is at $x=-3$.
$x=-3$. The vertex $x$-coordinate is at $x=-3$.
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What is the minimum value of $f(x)=x^2+10x+16$?
What is the minimum value of $f(x)=x^2+10x+16$?
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$-9$. The $k$ value in vertex form gives the extremum.
$-9$. The $k$ value in vertex form gives the extremum.
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What is the vertex form of $f(x)=x^2-12x+20$ after completing the square?
What is the vertex form of $f(x)=x^2-12x+20$ after completing the square?
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$f(x)=(x-6)^2-16$. Complete the square: $(x-6)^2-36+20=(x-6)^2-16$.
$f(x)=(x-6)^2-16$. Complete the square: $(x-6)^2-36+20=(x-6)^2-16$.
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What is the minimum value of $f(x)=x^2-12x+20$?
What is the minimum value of $f(x)=x^2-12x+20$?
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$-16$. The $k$ value in vertex form gives the extremum.
$-16$. The $k$ value in vertex form gives the extremum.
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What is the vertex form of $f(x)=-x^2+2x+8$ after completing the square?
What is the vertex form of $f(x)=-x^2+2x+8$ after completing the square?
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$f(x)=-(x-1)^2+9$. Factor out $-1$: $-(x-1)^2+1+8=-(x-1)^2+9$.
$f(x)=-(x-1)^2+9$. Factor out $-1$: $-(x-1)^2+1+8=-(x-1)^2+9$.
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What is the maximum value of $f(x)=-x^2+2x+8$?
What is the maximum value of $f(x)=-x^2+2x+8$?
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$9$. The $k$ value in vertex form gives the extremum.
$9$. The $k$ value in vertex form gives the extremum.
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What is the vertex form of $f(x)=4x^2-4x+1$ after completing the square?
What is the vertex form of $f(x)=4x^2-4x+1$ after completing the square?
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$f(x)=4\left(x-\frac{1}{2}\right)^2$. Factor out $4$: $4\left[\left(x-\frac{1}{2}\right)^2-\frac{1}{4}\right]+1=4\left(x-\frac{1}{2}\right)^2$.
$f(x)=4\left(x-\frac{1}{2}\right)^2$. Factor out $4$: $4\left[\left(x-\frac{1}{2}\right)^2-\frac{1}{4}\right]+1=4\left(x-\frac{1}{2}\right)^2$.
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What is the minimum value of $f(x)=4x^2-4x+1$?
What is the minimum value of $f(x)=4x^2-4x+1$?
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$0$. The $k$ value in vertex form gives the extremum.
$0$. The $k$ value in vertex form gives the extremum.
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What is the vertex form of $f(x)=2x^2-6x-8$ after completing the square?
What is the vertex form of $f(x)=2x^2-6x-8$ after completing the square?
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$f(x)=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}$. Factor out $2$: $2\left[\left(x-\frac{3}{2}\right)^2-\frac{9}{4}\right]-8=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}$.
$f(x)=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}$. Factor out $2$: $2\left[\left(x-\frac{3}{2}\right)^2-\frac{9}{4}\right]-8=2\left(x-\frac{3}{2}\right)^2-\frac{25}{2}$.
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What is the minimum value of $f(x)=2x^2-6x-8$?
What is the minimum value of $f(x)=2x^2-6x-8$?
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$-\frac{25}{2}$. The $k$ value in vertex form gives the extremum.
$-\frac{25}{2}$. The $k$ value in vertex form gives the extremum.
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What is the vertex form of $f(x)=-3x^2-12x+6$ after completing the square?
What is the vertex form of $f(x)=-3x^2-12x+6$ after completing the square?
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$f(x)=-3(x+2)^2+18$. Factor out $-3$: $-3[(x+2)^2-4]+6=-3(x+2)^2+18$.
$f(x)=-3(x+2)^2+18$. Factor out $-3$: $-3[(x+2)^2-4]+6=-3(x+2)^2+18$.
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What is the maximum value of $f(x)=-3x^2-12x+6$?
What is the maximum value of $f(x)=-3x^2-12x+6$?
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$18$. The $k$ value in vertex form gives the extremum.
$18$. The $k$ value in vertex form gives the extremum.
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What is the maximum value of $f(x)=a(x-h)^2+k$ when $a<0$?
What is the maximum value of $f(x)=a(x-h)^2+k$ when $a<0$?
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$k$. The $y$-coordinate of the vertex is the maximum.
$k$. The $y$-coordinate of the vertex is the maximum.
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What is the minimum value of $f(x)=a(x-h)^2+k$ when $a>0$?
What is the minimum value of $f(x)=a(x-h)^2+k$ when $a>0$?
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$k$. The $y$-coordinate of the vertex is the minimum.
$k$. The $y$-coordinate of the vertex is the minimum.
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What is the minimum value of $f(x)=\frac{1}{2}x^2+3x+1$?
What is the minimum value of $f(x)=\frac{1}{2}x^2+3x+1$?
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$-\frac{7}{2}$. The $k$ value in vertex form gives the extremum.
$-\frac{7}{2}$. The $k$ value in vertex form gives the extremum.
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What is the vertex form of $f(x)=x^2+10x+16$ after completing the square?
What is the vertex form of $f(x)=x^2+10x+16$ after completing the square?
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$f(x)=(x+5)^2-9$. Complete the square: $(x+5)^2-25+16=(x+5)^2-9$.
$f(x)=(x+5)^2-9$. Complete the square: $(x+5)^2-25+16=(x+5)^2-9$.
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What condition on $a$ makes the vertex of $f(x)=a(x-h)^2+k$ a maximum?
What condition on $a$ makes the vertex of $f(x)=a(x-h)^2+k$ a maximum?
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$a<0$. Negative $a$ makes the parabola open downward.
$a<0$. Negative $a$ makes the parabola open downward.
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What condition on $a$ makes the vertex of $f(x)=a(x-h)^2+k$ a minimum?
What condition on $a$ makes the vertex of $f(x)=a(x-h)^2+k$ a minimum?
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$a>0$. Positive $a$ makes the parabola open upward.
$a>0$. Positive $a$ makes the parabola open upward.
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What is the minimum value of $f(x)=x^2+6x+5$ found by completing the square?
What is the minimum value of $f(x)=x^2+6x+5$ found by completing the square?
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$-4$. The $k$ value in vertex form gives the extremum.
$-4$. The $k$ value in vertex form gives the extremum.
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What is the maximum value of $f(x)=-x^2-4x+1$?
What is the maximum value of $f(x)=-x^2-4x+1$?
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$5$. The $k$ value in vertex form gives the extremum.
$5$. The $k$ value in vertex form gives the extremum.
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What is the completed-square form of $x^2+bx$ (no constant term) ?
What is the completed-square form of $x^2+bx$ (no constant term) ?
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$\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4}$. Add and subtract $\left(\frac{b}{2}\right)^2$ to complete the square.
$\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4}$. Add and subtract $\left(\frac{b}{2}\right)^2$ to complete the square.
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What is the minimum value of $f(x)=x^2+bx$ expressed in terms of $b$?
What is the minimum value of $f(x)=x^2+bx$ expressed in terms of $b$?
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$-\frac{b^2}{4}$. The extremum occurs when $(x+\frac{b}{2})^2=0$.
$-\frac{b^2}{4}$. The extremum occurs when $(x+\frac{b}{2})^2=0$.
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What is the vertex $y$-coordinate for $f(x)=ax^2+bx+c$ using substitution?
What is the vertex $y$-coordinate for $f(x)=ax^2+bx+c$ using substitution?
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$f\left(-\frac{b}{2a}\right)$. Substitute the vertex $x$-coordinate into the function.
$f\left(-\frac{b}{2a}\right)$. Substitute the vertex $x$-coordinate into the function.
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What is the standard completed-square (vertex) form of a quadratic function?
What is the standard completed-square (vertex) form of a quadratic function?
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$f(x)=a(x-h)^2+k$. This is the standard form with vertex at $(h,k)$.
$f(x)=a(x-h)^2+k$. This is the standard form with vertex at $(h,k)$.
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What is the vertex of $f(x)=a(x-h)^2+k$ written in vertex form?
What is the vertex of $f(x)=a(x-h)^2+k$ written in vertex form?
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$(h,k)$. The vertex coordinates are directly read from the form.
$(h,k)$. The vertex coordinates are directly read from the form.
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What is the vertex form of $f(x)=x^2+6x+5$ after completing the square?
What is the vertex form of $f(x)=x^2+6x+5$ after completing the square?
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$f(x)=(x+3)^2-4$. Complete the square: $(x+3)^2-9+5=(x+3)^2-4$.
$f(x)=(x+3)^2-4$. Complete the square: $(x+3)^2-9+5=(x+3)^2-4$.
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What is the vertex form of $f(x)=x^2-8x+7$ after completing the square?
What is the vertex form of $f(x)=x^2-8x+7$ after completing the square?
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$f(x)=(x-4)^2-9$. Complete the square: $(x-4)^2-16+7=(x-4)^2-9$.
$f(x)=(x-4)^2-9$. Complete the square: $(x-4)^2-16+7=(x-4)^2-9$.
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What is the minimum value of $f(x)=x^2-8x+7$?
What is the minimum value of $f(x)=x^2-8x+7$?
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$-9$. The $k$ value in vertex form gives the extremum.
$-9$. The $k$ value in vertex form gives the extremum.
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At what $x$ does $f(x)=x^2-8x+7$ attain its minimum value?
At what $x$ does $f(x)=x^2-8x+7$ attain its minimum value?
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$x=4$. The vertex $x$-coordinate is at $x=4$.
$x=4$. The vertex $x$-coordinate is at $x=4$.
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What is the vertex form of $f(x)=x^2+4x+10$ after completing the square?
What is the vertex form of $f(x)=x^2+4x+10$ after completing the square?
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$f(x)=(x+2)^2+6$. Complete the square: $(x+2)^2-4+10=(x+2)^2+6$.
$f(x)=(x+2)^2+6$. Complete the square: $(x+2)^2-4+10=(x+2)^2+6$.
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