Question

State whether the graph of each quadratic function opens up or down. Then state whether the function has a maximum or minimum value.

$\begin{array}{l}a.f\left(x\right)=3x^2+4x-5\\ b.f\left(x\right)=-2x^2+9\\ c.f\left(x\right)=-5x^2-8x+2\\ d.f\left(x\right)=6x^2-5x\end{array}$

Short answer

1. The graph of function $f\left(x\right)=3x^2+4x-5$ opens up and has a minimum value.
2. The graph of function $f\left(x\right)=-2x^2+9$ opens down and has a maximum value.
3. The graph of function $f\left(x\right)=-5x^2-8x+2$ opens down and has a maximum value.

d. The graph of function $f\left(x\right)=6x^2-5x$ opens up and has a minimum value.

Step-by-step solution

aStep 1. Use the concept.

The graph of $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

opens up and has a minimum value when $a>0$, and

opens down and has a maximum value when $a<0$

Step 2. Given Information.

The given function is $f\left(x\right)=3x^2+4x-5$

Step 3. Solution

In the function $f\left(x\right)=3x^2+4x-5$, we have $a=3>0$

So, the graph of function $f\left(x\right)=3x^2+4x-5$ opens up and has a minimum value.

bStep 1. Use the concept.

The graph of $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

opens up and has a minimum value when $a>0$, and

opens down and has a maximum value when $a<0$

Step 2. Given Information.

The given function is $f\left(x\right)=-2x^2+9$

Step 3. Solution.

In the function$f\left(x\right)=-2x^2+9$, we have $a=-2<0$

So, the graph of function $f\left(x\right)=-2x^2+9$ opens down and has a maximum value.

cStep 1. Use the concept.

The graph of $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

opens up and has a minimum value when $a>0$, and

opens down and has a maximum value when $a<0$

Step 2. Given Information.

The given function is $f\left(x\right)=-5x^2-8x+2$

Step 3. Solution.

In the function $f\left(x\right)=-5x^2-8x+2$, we have $a=-5<0$

So, the graph of function $f\left(x\right)=-5x^2-8x+2$ opens down and has a maximum value.

dStep 1. Use the concept.

The graph of $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

opens up and has a minimum value when $a>0$, and

opens down and has a maximum value when $a<0$

Step 2. Given Information.

The given function is $f\left(x\right)=6x^2-5x$

Step 3. Solution.

In the function $f\left(x\right)=6x^2-5x$, we have $a=6>0$

So, the graph of function $f\left(x\right)=6x^2-5x$ opens up and has a minimum value.