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Access these online resources for additional instruction and practice with quadratic equations.
general form of a quadratic function | $$f(x)=a{x}^{2}+bx+c$$ |
standard form of a quadratic function | $$f(x)=a{(x-h)}^{2}+k$$ |
Explain the advantage of writing a quadratic function in standard form.
When written in that form, the vertex can be easily identified.
How can the vertex of a parabola be used in solving real-world problems?
Explain why the condition of $\text{\hspace{0.17em}}a\ne 0\text{\hspace{0.17em}}$ is imposed in the definition of the quadratic function.
If $\text{\hspace{0.17em}}a=0\text{\hspace{0.17em}}$ then the function becomes a linear function.
What is another name for the standard form of a quadratic function?
What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?
If possible, we can use factoring. Otherwise, we can use the quadratic formula.
For the following exercises, rewrite the quadratic functions in standard form and give the vertex.
$f\left(x\right)={x}^{2}-12x+32$
$g\left(x\right)={x}^{2}+2x-3$
$f(x)={(x+1)}^{2}-2,\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(-1,-4\right)$
$f(x)={x}^{2}-x$
$f(x)={x}^{2}+5x-2$
$f(x)={\left(x+\frac{5}{2}\right)}^{2}-\frac{33}{4},\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(-\frac{5}{2},-\frac{33}{4}\right)$
$h\left(x\right)=2{x}^{2}+8x-10$
$k\left(x\right)=3{x}^{2}-6x-9$
$f(x)=3{(x-1)}^{2}-12,\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}(1,-12)$
$f(x)=2{x}^{2}-6x$
$f(x)=3{x}^{2}-5x-1$
$f(x)=3{\left(x-\frac{5}{6}\right)}^{2}-\frac{37}{12},\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(\frac{5}{6},-\frac{37}{12}\right)$
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
$y\left(x\right)=2{x}^{2}+10x+12$
$f\left(x\right)=2{x}^{2}-10x+4$
Minimum is $\text{\hspace{0.17em}}-\frac{17}{2}\text{\hspace{0.17em}}$ and occurs at $\text{\hspace{0.17em}}\frac{5}{2}.\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}x=\frac{5}{2}.$
$f(x)=-{x}^{2}+4x+3$
$f(x)=4{x}^{2}+x-1$
Minimum is $\text{\hspace{0.17em}}-\frac{17}{16}\text{\hspace{0.17em}}$ and occurs at $\text{\hspace{0.17em}}-\frac{1}{8}.\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}x=-\frac{1}{8}.$
$h\left(t\right)=\mathrm{-4}{t}^{2}+6t-1$
$f(x)=\frac{1}{2}{x}^{2}+3x+1$
Minimum is $\text{\hspace{0.17em}}-\frac{7}{2}\text{\hspace{0.17em}}$ and occurs at $\text{\hspace{0.17em}}\mathrm{-3.}\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}x=\mathrm{-3.}$
$f(x)=-\frac{1}{3}{x}^{2}-2x+3$
For the following exercises, determine the domain and range of the quadratic function.
$f(x)={(x-3)}^{2}+2$
Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}[2,\infty ).$
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