Chapter 3.3, Problem 50AYU

In Problems 49-54, determine the quadratic function whose graph is given.

To determine

To calculate: The quadratic function using the given graphs.

Answer to Problem 50AYU

The equation of the given graph is $f(x)=-x^2+4x-1$.

Explanation of Solution

Given:

The given graph is

Formula Used:

The general form of a quadratic equation is $f(x)=a{(x-h)}^2+k,a\ne 0$.

This general equation can also be written as $f(x)=a{x}^2+bx+c,a\ne 0$, where

$\begin{array}{l}h=-\frac{b}{2a}\\ \text{and}\\ k=f\left(-\frac{b}{2a}\right)\end{array}$

If $a>0$, the graph opens upwards.

If $a\text{ < }0$, the graph opens downwards.

The vertex of the above function is $(h,k)=\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)$.

The axis of symmetry will be $x=\frac{-b}{2a}$.

We can find the $y\text{-intercept}$ by equating the equation at $x=0$.

We can find the $x\text{-intercept}$ by equation the equation at $y=0$.

1. If $b^2-4ac=0$ then the vertex is the $x\text{-intercept}$.

2. If $b^2-4ac\text{ < }0$ then the graph has no $x\text{-intercept}$.

3. If $b^2-4ac\text{ > }0$ then the vertex is the $x\text{-intercept}$.

Calculation:

Here, we can see that the graph is open downwards. Therefore, we get $a<0$.

The vertex of the given graph is at $(2,3)$.

Thus, we have

$\begin{array}{l}(h,k)=\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)=(2,3)\\ \Rightarrow h=2\\ \text{and}\\ k=3\end{array}$

Therefore, the equation of the given graph is

$\begin{array}{l}f(x)=a{\left(x-(2)\right)}^2+(3)\\ =a{\left(x-2\right)}^2+3\end{array}$

Now, we have to determine the value of $a$.

From the given graph, we can see that the

From the given graph, we can see that the $y\text{-intercept}$ is at $f(0)=-1$.

Therefore, we have

$\begin{array}{l}f(0)=-1\\ \Rightarrow -1=a{\left(0-2\right)}^2+3\\ \Rightarrow -1=4a+3\\ \Rightarrow a=-1\end{array}$

Therefore, we get $a=-1$.

Thus, the equation of the given graph is

$\begin{array}{l}f(x)=-1{\left(x-2\right)}^2+3\\ =-\left(x^2+4-4x\right)+3\\ =-x^2+4x-1\end{array}$

The equation is $f(x)=-x^2+4x-1$.