# 5.1 Quadratic functions  (Page 3/15)

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A coordinate grid has been superimposed over the quadratic path of a basketball in [link] . Find an equation for the path of the ball. Does the shooter make the basket?

The path passes through the origin and has vertex at so $\text{\hspace{0.17em}}\left(h\right)x=–\frac{7}{16}{\left(x+4\right)}^{2}+7.\text{\hspace{0.17em}}$ To make the shot, $\text{\hspace{0.17em}}h\left(-7.5\right)\text{\hspace{0.17em}}$ would need to be about 4 but $\text{\hspace{0.17em}}h\left(–7.5\right)\approx 1.64;\text{\hspace{0.17em}}$ he doesn’t make it.

Given a quadratic function in general form, find the vertex of the parabola.

1. Identify
2. Find $\text{\hspace{0.17em}}h,\text{\hspace{0.17em}}$ the x -coordinate of the vertex, by substituting $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ into $\text{\hspace{0.17em}}h=–\frac{b}{2a}.$
3. Find $\text{\hspace{0.17em}}k,\text{\hspace{0.17em}}$ the y -coordinate of the vertex, by evaluating $\text{\hspace{0.17em}}k=f\left(h\right)=f\left(-\frac{b}{2a}\right).$

## Finding the vertex of a quadratic function

Find the vertex of the quadratic function $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}–6x+7.\text{\hspace{0.17em}}$ Rewrite the quadratic in standard form (vertex form).

$\begin{array}{c}\text{The horizontal coordinate of the vertex will be at}\hfill \\ & \hfill h& =& -\frac{b}{2a}\hfill \\ & & =& \frac{-6}{2\left(2\right)}\hfill \\ & & =& \frac{6}{4}\hfill \\ & & =& \frac{3}{2}\hfill \\ \text{The vertical coordinate of the vertex will be at}\hfill \\ & \hfill k& =& f\left(h\right)\hfill \\ & & =& f\left(\frac{3}{2}\right)\hfill \\ & & =& 2{\left(\frac{3}{2}\right)}^{2}-6\left(\frac{3}{2}\right)+7\hfill \\ & & =& \frac{5}{2}\hfill \end{array}$

Rewriting into standard form, the stretch factor will be the same as the $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the $\mathrm{“a”}$ from the general form.

$\begin{array}{ccc}\hfill f\left(x\right)& =& a{x}^{2}+bx+c\hfill \\ \hfill f\left(x\right)& =& 2{x}^{2}-6x+7\hfill \end{array}$

The standard form of a quadratic function prior to writing the function then becomes the following:

$f\left(x\right)=2{\left(x–\frac{3}{2}\right)}^{2}+\frac{5}{2}$

Given the equation $\text{\hspace{0.17em}}g\left(x\right)=13+{x}^{2}-6x,$ write the equation in general form and then in standard form.

$g\left(x\right)={x}^{2}-6x+13\text{\hspace{0.17em}}$ in general form; $\text{\hspace{0.17em}}g\left(x\right)={\left(x-3\right)}^{2}+4\text{\hspace{0.17em}}$ in standard form

## Finding the domain and range of a quadratic function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y -values greater than or equal to the y -coordinate at the turning point or less than or equal to the y -coordinate at the turning point, depending on whether the parabola opens up or down.

## Domain and range of a quadratic function

The domain of any quadratic function is all real numbers unless the context of the function presents some restrictions.

The range of a quadratic function written in general form $\text{\hspace{0.17em}}f\left(x\right)=a{x}^{2}+bx+c\text{\hspace{0.17em}}$ with a positive $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ value is $\text{\hspace{0.17em}}f\left(x\right)\ge f\left(-\frac{b}{2a}\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left[f\left(-\frac{b}{2a}\right),\infty \right);\text{\hspace{0.17em}}$ the range of a quadratic function written in general form with a negative $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ value is $\text{\hspace{0.17em}}f\left(x\right)\le f\left(-\frac{b}{2a}\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-\infty ,f\left(-\frac{b}{2a}\right)\right].$

The range of a quadratic function written in standard form $\text{\hspace{0.17em}}f\left(x\right)=a{\left(x-h\right)}^{2}+k\text{\hspace{0.17em}}$ with a positive $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ value is $\text{\hspace{0.17em}}f\left(x\right)\ge k;\text{\hspace{0.17em}}$ the range of a quadratic function written in standard form with a negative $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ value is $\text{\hspace{0.17em}}f\left(x\right)\le k.$

Given a quadratic function, find the domain and range.

1. Identify the domain of any quadratic function as all real numbers.
2. Determine whether $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is positive or negative. If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is positive, the parabola has a minimum. If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is negative, the parabola has a maximum.
3. Determine the maximum or minimum value of the parabola, $\text{\hspace{0.17em}}k.$
4. If the parabola has a minimum, the range is given by $\text{\hspace{0.17em}}f\left(x\right)\ge k,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left[k,\infty \right).\text{\hspace{0.17em}}$ If the parabola has a maximum, the range is given by $\text{\hspace{0.17em}}f\left(x\right)\le k,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-\infty ,k\right].$

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