Chapter 1.1, Problem 43E

Solution

To find: The equations of quadratic functions whose graph have a given axis of symmetry.

The quadratic functions are: $y\left(x\right)=\frac{1}{3}{\left(x-4\right)}^2+1$ , $y\left(x\right)=\frac{1}{2}{\left(x-4\right)}^2+1$ , and $y\left(x\right)=\frac{3}{4}{\left(x-4\right)}^2+1$ .

Given data: The axis of symmetry of parabolas is $x=4$ .

Method/Formula used:

A quadratic function $y\left(x\right)=a{\left(x-h\right)}^2+k$ represents a parabola with $x=4$ as the axis of symmetry, $\left(h,k\right)$ the vertex and ‘ a’ normal to the focus to that parabola.

Calculation:

Three different quadratic functions whose graph has $x=4$ as the axis of symmetry can be obtained by choosing constant $\left(h,k\right)$ and variable ‘ a’ .

These are:

1- Choose $h=4$ , $k=1$ , and $a=1$ , the quadratic function is $y\left(x\right)=\frac{1}{3}{\left(x-4\right)}^2+1$ .

2- Choose $h=4$ , $k=1$ , and $a=2$ , the quadratic function is $y\left(x\right)=\frac{1}{2}{\left(x-4\right)}^2+1$ .

3- Choose $h=4$ , $k=1$ , and $a=3$ , the quadratic function is $y\left(x\right)=\frac{3}{4}{\left(x-4\right)}^2+1$ .

The graphs of all the three functions are shown in single window, Fig. (1).

The intercepts on the y-axis are: 0.633, 9 and 13 respectively. The axis of symmetry and the vertex of all the function is $x=4$ and $\left(4,1\right)$ .