Chapter 2.2, Problem 10E

To determine

To identify (match):The graph of a polynomial from the given list of graphs.

Answer to Problem 10E

The polynomial function $f\left(x\right)=x^2-4x$ represents the graph (h) from the list.

Explanation of Solution

Given information:

The given polynomial function is $f\left(x\right)=x^2-4x$ .

Concept used:

The graph of standard form of quadratic function $f\left(x\right)=a{\left(x-h\right)}^2+k$ is a parabola with vertex $\left(h,k\right)$ . If $a>0$ the parabola is concave-upwards.

The x- intercept of the graph of quadratic function $f\left(x\right)$ is $\left(a,0\right)$ , where $f\left(a\right)=0$ .

The given quadratic polynomial $f\left(x\right)=x^2-4x$ can be written as:

  $f\left(x\right)=1\cdot \left(x^2-4x+4\right)-4$

  $\Rightarrow f\left(x\right)=1\cdot {\left(x-2\right)}^2-4$ ...............(1)

The vertex of the parabola represented by $f\left(x\right)=x^2-4x$ is $\left(2,-4\right)$ . Since, $a=1>0$ so the parabola opening upwards.

Now, $f\left(x\right)=0\Rightarrow x^2-4x=0$

  $\Rightarrow x\left(x-4\right)=0\Rightarrow x=0,x=4$

Hence, the x-intercepts of the graph of $f\left(x\right)=x^2-4x$ are $\left(0,0\right)$ and $\left(4,0\right)$ .

We see that the graph (h) is a parabola with vertex $\left(2,-4\right)$ that opens upwards and has $\left(0,0\right)$ and $\left(4,0\right)$ as its x-intercepts.

  

Hence, the quadratic polynomial function $f\left(x\right)=x^2-4x$ represents the graph (h) from the list.

Conclusion:

The quadratic polynomial function $f\left(x\right)=x^2-4x$ represents the graph (h) from the list.