Chapter SH, Problem 4.2EP

a.

To determine

Find the y-intercept, axis of symmetry and the x-coordinate of the vertex.

Answer to Problem 4.2EP

The y-intercept $=0$

Theequation of the axis of symmetry is $x=\frac{5}{4}$

The x-coordinate of the vertex $=\frac{5}{4}$

Explanation of Solution

Given:

The given function is $f\left(x\right)=2x^2-5x$ .

Calculation:

  $f\left(x\right)=2x^2-5x$

The function is of the form $f\left(x\right)=a{x}^2+bx+c$ .

  $a=2,b=-5,c=0$

y-intercept $=c=0$ .

Equation of the axis of symmetry.

  $\begin{array}{l}x=-\frac{b}{2a}\\ \Rightarrow x=-\frac{\left(-5\right)}{2\cdot 2}\\ \Rightarrow x=\frac{5}{4}\end{array}$

The axis of symmetry intersects a parabola at only one point, called the vertex.

The x-coordinate of the vertex $=-\frac{b}{2a}$

  $=-\frac{\left(-5\right)}{2\cdot 2}$

  $=\frac{5}{4}$

Hence

the y-intercept $=0$

theequation of the axis of symmetry is $x=\frac{5}{4}$

the x-coordinate of the vertex $=\frac{5}{4}$

b.

To determine

Create a table for the values with vertex.

Answer to Problem 4.2EP

The table for the values of $y$ with respect to $x$ .

$x$	$y=2x^2-5x$	$\left(x,y\right)$
$\begin{array}{l}-2\\ -1\\ 0\\ 1.25\\ 3.5\\ 4.5\end{array}$	$\begin{array}{l}18\\ 7\\ 0\\ -3.125\\ 7\\ 18\end{array}$	$\begin{array}{l}\left(-2,18\right)\\ \left(-1,7\right)\\ \left(0,0\right)\\ \left(1.25,-3.125\right)\\ \left(3.5,7\right)\\ \left(4.5,18\right)\end{array}$

Explanation of Solution

Given:

The given function is $f\left(x\right)=2x^2-5x$ .

Calculation:

  $f\left(x\right)=2x^2-5x$

The function is of the form $f\left(x\right)=a{x}^2+bx+c$ .

  $a=2,b=-5,c=0$

The x-coordinate of the vertex $=-\frac{b}{2a}$

  $=-\frac{\left(-5\right)}{2\cdot 2}$

  $=\frac{5}{4}$

Substitute the value of $x=\frac{5}{4}$ in the given function.

  $\begin{array}{l}y=2{\left(\frac{5}{4}\right)}^2-5\left(\frac{5}{4}\right)\\ =2\cdot \frac{25}{16}-\frac{25}{4}\\ =\frac{25}{8}-\frac{25}{4}\\ =-\frac{25}{8}\end{array}$

Vertex $=\left(\frac{5}{4},-\frac{25}{8}\right)$

The table for the values of $y$ with respect to $x$ is given below.

$x$	$y=2x^2-5x$	$\left(x,y\right)$
$\begin{array}{l}-2\\ -1\\ 0\\ 1.25\\ 3.5\\ 4.5\end{array}$	$\begin{array}{l}18\\ 7\\ 0\\ -3.125\\ 7\\ 18\end{array}$	$\begin{array}{l}\left(-2,18\right)\\ \left(-1,7\right)\\ \left(0,0\right)\\ \left(1.25,-3.125\right)\\ \left(3.5,7\right)\\ \left(4.5,18\right)\end{array}$

c.

To determine

Draw a graph for the given function.

Explanation of Solution

Given:

The given function is $f\left(x\right)=2x^2-5x$ .

Calculation:

  $f\left(x\right)=2x^2-5x$

The function is of the form $f\left(x\right)=a{x}^2+bx+c$ .

  $a=2,b=-5,c=0$

y-intercept $=c=0$ .

Equation of the axis of symmetry.

  $\begin{array}{l}x=-\frac{b}{2a}\\ \Rightarrow x=-\frac{\left(-5\right)}{2\cdot 2}\\ \Rightarrow x=\frac{5}{4}\end{array}$

The axis of symmetry intersects a parabola at only one point, called the vertex.

The x-coordinate of the vertex $=-\frac{b}{2a}$

  $=-\frac{\left(-5\right)}{2\cdot 2}$

  $=\frac{5}{4}$

Draw a table for the values of $y$ with respect to $x$ .

$x$	$y=2x^2-5x$	$\left(x,y\right)$
$\begin{array}{l}-2\\ -1\\ 0\\ 1.25\\ 3.5\\ 4.5\end{array}$	$\begin{array}{l}18\\ 7\\ 0\\ -3.125\\ 7\\ 18\end{array}$	$\begin{array}{l}\left(-2,18\right)\\ \left(-1,7\right)\\ \left(0,0\right)\\ \left(1.25,-3.125\right)\\ \left(3.5,7\right)\\ \left(4.5,18\right)\end{array}$

Hence the graph of the function is given below.