Q67. - Algebra 1

Question

MULTIPLE REPRESENTATIONS in this problem, you will investigate solving quadratic equations using tables.

a. ALGEBRAIC Determine the related function for each equation. Copy and complete the table below.

b. GRAPHICAL Graph each related function with a graphing calculator.

c. ANALYTICAL use the table feature on your calculator to determine the zeros of each related function. Record the zeros in the table above. Also, record the values of the function one unit less than and one unit more than each zero.

d. VERBAL Compare the signs of the function values for x values just before and just after a zero. What happens to the sign of the function value before and after a zero?

Step-by-Step Solution

Answer

a. 

Equation	Related function
$x^2-x=12$	$y=x^2-x-\mathbf{12}$
$x^2+8x=9$	$y=x^2+\mathbf{8}x-\mathbf{9}$
$x^2=14x-24$	$y=x^2-\mathbf{14}x+\mathbf{24}$
$x^2+16x=-28$	$y=x^2+\mathbf{16}x+\mathbf{28}$

 

b. 

 1. 

2. 

3.

4.

c. 

 

Equation	Related function	Zeros	y-values
$x^2-x=12$	$y=x^2-x-12$	$x=-\mathbf{3}\mathbf{,}\mathbf{4}$	$$\begin{gathered} -\mathbf{3}\mathbf{:}\mathbf{8}\mathbf{,}-\mathbf{6} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{4}\mathbf{:}\mathbf{-}\mathbf{6}\mathbf{,}\mathbf{8}\mathbf{ } \end{gathered}$$
$x^2+8x=9$	$y=x^2+8x-9$	$x=-\mathbf{9}\mathbf{,}\mathbf{1}$	$$\begin{gathered} \mathbf{-}\mathbf{9}\mathbf{:}\mathbf{11}\mathbf{,}\mathbf{-}\mathbf{9} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{1}\mathbf{:}-\mathbf{9}\mathbf{,}\mathbf{11} \end{gathered}$$
$x^2=14x-24$	$y=x^2-14x+24$	$x=\mathbf{2}\mathbf{,}\mathbf{12}$	$\begin{array}{c}\mathbf{2}\mathbf{:}\mathbf{11}\mathbf{,}-\mathbf{9}\\ \mathbf{12}\mathbf{:}-\mathbf{9}\mathbf{,}\mathbf{11}\end{array}$
$x^2+16x=-28$	$y=x^2+16x+28$	$x=-\mathbf{14}\mathbf{,}-\mathbf{2}$	$\begin{array}{c}-\mathbf{14}\mathbf{:}\mathbf{13}\mathbf{,}-\mathbf{11}\\ -\mathbf{2}\mathbf{:}-\mathbf{11}\mathbf{,}\mathbf{13}\end{array}$

 

d. The function values have opposite signs just before and just after the zeros of the function.

Part a. Step 1. Find the related functions.

To find the related functions, write the given equations in standard form, $y=a{x}^2+bx+c$.

 

$x^2-x=12$ can be written in related functions as $y=x^2-x-12$.

$x^2+8x=9$ can be written in related functions as $y=x^2+8x-9$.

$x^2=14x-24$ can be written in related functions as $y=x^2-14x+24$.

$x^2+16x=-28$ can be written in related functions as $y=x^2+16x+28$.

Part a. Step 2. Make a table.

Make a table of related functions for each equation.

 

Equation	Related function
$x^2-x=12$	$y=x^2-x-12$
$x^2+8x=9$	$y=x^2+8x-9$
$x^2=14x-24$	$y=x^2-14x+24$
$x^2+16x=-28$	$y=x^2+16x+28$

 

Part b. Step 1. Graph.

Graph $y=x^2-x-12$ using a graphing calculator.

Part b. Step 2. Graph.

Graph $y=x^2+8x-9$ using a graphing calculator.

Part b. Step 3. Graph.

Graph $y=x^2-14x+24$ using a graphing calculator.

Part b. Step 4. Graph.

Graph $y=x^2+16x+28$ using a graphing calculator.

Part c. Step 1. Determine the zeros of the function.

Use the table function on the calculator to identify the zeros. The zeros are the values where y is 0.

 

Equation	Related function	Zeros
$x^2-x=12$	$y=x^2-x-12$	$x=-3,4$
$x^2+8x=9$	$y=x^2+8x-9$	$x=-9,1$
$x^2=14x-24$	$y=x^2-14x+24$	$x=2,12$
$x^2+16x=-28$	$y=x^2+16x+28$	$x=-14,-2$

 

Part c. Step 2. Values of function.

Make a list of y-values that are to the left and right of the zeros.

 

Equation	Related function	Zeros	y-values
$x^2-x=12$	$y=x^2-x-12$	$x=-3,4$	$\begin{array}{c}-3:8,-6\\ 4:-6,8\end{array}$
$x^2+8x=9$	$y=x^2+8x-9$	$x=-9,1$	$\begin{array}{c}-9:11,-9\\ 1:-9,11\end{array}$
$x^2=14x-24$	$y=x^2-14x+24$	$x=2,12$	$\begin{array}{c}2:11,-9\\ 12:-9,11\end{array}$
$x^2+16x=-28$	$y=x^2+16x+28$	$x=-14,-2$	$\begin{array}{c}-14:13,-11\\ -2:-11,13\end{array}$

Part d. Step 1. Observe the table.

Observe the following table. Compare the signs of the function values for x values just before and just after a zero.

 

Function	Zeros	y-values
$y=x^2-x-12$	$x=-3,4$	$\begin{array}{c}-3:8,-6\\ 4:-6,8\end{array}$
$y=x^2+8x-9$	$x=-9,1$	$\begin{array}{c}-9:11,-9\\ 1:-9,11\end{array}$
$y=x^2-14x+24$	$x=2,12$	$\begin{array}{c} 2:11,-9\\ 12:-9,11\end{array}$
$y=x^2+16x+28$	$x=-14,-2$	$\begin{array}{c}-14:13,-11\\ -2:-11,13\end{array}$

 

Part d. Step 2. Write the conclusion.

The function values have opposite signs just before and just after the zeros of the function.