Q29. - Algebra 1

Question

Write an equation for the function shown in the graph.

Step-by-Step Solution

Answer

The required equation is ${\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{3}$. 

Step 1. Understand the concept used .

If $a{x}^2+bx+c=0$, with $a\ne 0$, then $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Solving using factoring can be done by splitting the middle term of the equation and then making groups and equating them to zero.

Step 2. Substitute the values .

From the graph, three points on the graph detected are: $(0,-3),(-1,-1),(1,-1)$.

Substitute the points $(0,-3),(-1,-1),(1,-1)$ on the equation $y=a{x}^2+bx+c$ to obtain three equations in variables $a, b$ and c. 

 

1. $(x,y)=\left(0,-3\right)$

$$\begin{gathered} y=a{x}^2+bx+c \\ -3=a{\left(0\right)}^2+b\left(0\right)+c \\ c=-3 \end{gathered}$$

2. $(x,y)=\left(-1,-1\right)$

$$\begin{gathered} -1=a{\left(-1\right)}^2+b(-1)-3 \\ -1+3=a-b \\ 2=a-b\left(2\right) \end{gathered}$$

 3. $(x,y)=\left(1,-1\right)$

$$\begin{gathered} -1=a{\left(1\right)}^2+b\left(1\right)-3 \\ -1+3=a+b \\ 2=a+b\left(3\right) \end{gathered}$$

Step 3. Solve the equations for a and b .

Add equations (2) and (3) to eliminate variable b and solve for a.

 $a+b+\left(a-b\right)=2+2$

                   $$\begin{gathered} 2a=4 \\ a=\frac{4}{2} \\ =2 \end{gathered}$$ 

Substitute 2 for an into equation (3) and solve for $b$.

$$\begin{gathered} a+b=2 \\ 2+b=2 \\ b=0 \end{gathered}$$

Since, $a=2;b=0;c=-3$, therefore, the required equation is $y=2x^2-3$.