Chapter 5, Problem 1RE

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts.Finally, graph the function.

1. $f\left(x\right)=x^2-4x-5$

To determine

Write the below quadratic function in standard form. Find the vertex and axes intercepts and graph the function

$f\left(x\right)=x^2-4x-5$

Answer to Problem 1RE

The standard form of quadratic equation is$f\left(x\right)={\left(x-2\right)}^2-9$. Vertex is$\left(2,-9\right)$, while x-intercepts are$\left(-1,0\right)$and$\left(5,0\right)$and y-intercept is$\left(0,-5\right)$

Explanation of Solution

Given:

Quadratic function$f\left(x\right)=x^2-4x-5$

Formula Used:

Let us consider a quadratic function in the form of$f\left(x\right)=a{x}^2+bx+c$, then the standard form of the equation is given as

$f\left(x\right)=a{\left(x-h\right)}^2+k$

Where$h=-\frac{b}{2a}$

$k=f\left(h\right)$

Calculation:

Quadratic function is given as

$f\left(x\right)=x^2-4x-5$

Comparing the above function with general quadratic function, we get

$a=1$

$b=-4$

$c=-5$

Thus, the value of$h$and$k$are calculated as follows:

$h=-\frac{\left(-4\right)}{2\times 1}$

$h=\frac{4}{2}$

$h=2$

And,$k=f\left(2\right)$

$k={\left(2\right)}^2-4\left(2\right)-5$

$k=4-8-5$

$k=-9$

Now, the standard form of quadratic function is given as

$f\left(x\right)=a{\left(x-h\right)}^2+k$

Substituting the values$a=1$,$h=2$and$k=-9$in above equation, we get

$f\left(x\right)=1{\left(x-\left(2\right)\right)}^2+\left(-9\right)$

$f\left(x\right)={\left(x-2\right)}^2-9$

Thus, the standard form of quadratic equation is$f\left(x\right)={\left(x-2\right)}^2-9$

Vertex is given as$\left(h,k\right)$

From above, the vertex is$\left(2,-9\right)$

Now, we need to sketch the graph.

From the graph,

x-intercepts are$\left(-1,0\right)$and$\left(5,0\right)$

y-intercept is$\left(0,-5\right)$

Conclusion:

Hence, the standard form of quadratic equation is$f\left(x\right)={\left(x-2\right)}^2-9$. Vertex is$\left(2,-9\right)$, while x-intercepts are$\left(-1,0\right)$and$\left(5,0\right)$and y-intercept is$\left(0,-5\right)$