Q. 81 - Precalculus Enhanced with Graphing Utilities

Question

Answer Problems 81 and 82 using the following: A quadratic function of the form $f\left(x\right)=a{x}^2+bx+c$ with $b^2-4ac>0$may also be written in the form $f\left(x\right)=a\left(x-r_1\right)\left(x-r_2\right)$ where $r_1 and r_2$ are the $x$-intercepts of the graph of the quadratic function.

(a) Find a quadratic function whose $x$-intercepts are $-3$ and $1$ with$a=1, a=2, a=-2, a=5.$

 (b) How does the value of $a$ affect the intercepts?

(c) How does the value of $a$ affect the axis of symmetry?

(d) How does the value of $a$affect the vertex?

(e) Compare the x-coordinate of the vertex with the midpoint of the $x$-intercepts. What might you conclude? 

Step-by-Step Solution

Answer

(a) The quadratic functions are,

$f\left(x\right)=x^2+2x-3.$

$f\left(x\right)=x^2+2x-3.$

$f\left(x\right)=2x^2+4x-6.$

$f\left(x\right)=-2x^2-4x+6.$

$f\left(x\right)=5x^2+10x-3.$

(b) The value $a$ does not affect intercept.

(c) The axis of symmetry remains the same.

(d) The vertex has the same coordinate $x$ but it shifts up or down for $2a.$

(e) The mid-point is the same.

Part (a) Step 1. Introduction

A quadratic function is written in $f\left(x\right)=a{x}^2+bx+c.$ We need to determine a quadratic function whose $x$-intercept is $-3$ and $1$ with $a=1, a=2, a=-2, a=5.$

Step 2. Simplify

$f\left(x\right)=a{x}^2+bx+c$ maybe written in the form $f\left(x\right)=a\left(x-r_1\right)\left(x-r_2\right),r_1,r_2 and x$-intercept.

When $a=1, f\left(x\right)=\left(x+3\right)\left(x-1\right).$

                             $f\left(x\right)=x^2+2x-3.$

When $a=2, f\left(x\right)=2\left(x+3\right)\left(x-1\right).$

                            $f\left(x\right)=2x^2+4x-6.$

When $a=-2, f\left(x\right)=-2\left(x+3\right)\left(x-1\right).$

                                $f\left(x\right)=-2x^2-4x+6.$

When $a=5, f\left(x\right)=5\left(x+3\right)\left(x-1\right).$

                             $f\left(x\right)=5x^2+10x-3.$

Part (b) Step 1. Introduction

An intercept is a point on the $y$- axis whereby the slope of a line passes we need to write how does the value $a$ affects the intercept.

Step 2. Conclusion

The value of $a$ does not affect interceptions.

Part (c) Introduction

An axis of symmetry is a line that divides an object into two equal halves thereby creating a mirror-like reflection of either side of the object we need to write how does the value of $a$ affect the axis of symmetry.

Step 2. Conclusion

The axis of symmetry remains the same.

Part (d) Step 1. Introduction

A vertex is an angular corner where two or more lines or edges meet. We need to write how does the value of $a$ affect vertex.

Step 2. Conclusion

The vertex has the same coordinate $x$ but it shifts up or down for $2a.$

Part (e) Step 1. Introduction

A mid-point is the mid-point of the signal. We need to compare the $x$- coordinate of the vertex with the mid-point of the $x$- intercept.

Step 2. Conclusion

The mid-point is the same.