a Substitute the given percentage. Then, rewrite the equation in standard form.
B
b Find the maximum value of the function. Is this value acceptable for this information?
A
a In 1993 and 2023
B
b No, see solution.
Practice makes perfect
a We are given a quadratic function that estimates the percent of U.S. households with high-speed Internet n years after 1990.
h= - 0.2 n^2 + 7.2 n +1.5
To determine when 20 % of the U.S. population will have high-speed Internet, we need to substitute 20 for h into the given quadratic function.
20= - 0.2 n^2 + 7.2 n +1.5
Let's rewrite the equation in standard form.
Now, we will solve it by using the Quadratic Formula. We first need to identify the values of a, b, and c.
- 0.2 n^2 + 7.2 n -18.5=0
⇕
- 0.2n^2+ 7.2n+( - 18.5)=0
We see that a= - 0.2, b= 7.2, and c= - 18.5. Let's substitute these values into the Quadratic Formula.
The solutions for this equation are n= - 7.2 ± sqrt(37.04)- 0.4. Let's separate them into the positive and negative cases.
n=- 7.2 ± sqrt(37.04)/- 0.4
n_1=- 7.2 + sqrt(37.04)/- 0.4
n_2=- 7.2 - sqrt(37.04)/- 0.4
n_1=7.2/0.4 - sqrt(37.04)/0.4
n_2=7.2/0.4 +sqrt(37.04)/0.4
n_1 ≈ 3
n_2 ≈ 33
Using the Quadratic Formula, we found that the solutions of the given equation are n_1≈ 3 and x_2≈33. Since n is the number of years since 1990, in 1993 and 2023, 20 % of the U.S. population will have high-speed Internet.
1990 + 3 = 1993
1990+33 =2023
b Since we are given a quadratic equation with a negative leading coefficient, the function has an maximum point.
h= - 0.2 n^2 + 7.2 n +1.5
Let's find this point to check if this quadratic equation is a good model. To do so, we need to identify a-, b-, and c-values of the related quadratic function.
f(x)= - 0.2 x^2 + 7.2 x + 1.5
We see that a= - 0.2, b= 7.2, and c= 1.5.
Recall that the maximum point is the vertex. We can write the expression for the vertex by stating the x- and f-coordinates in terms of a and b.
Vertex: ( - b/2 a, f( - b/2 a) )
We can now find the x-coordinate of the vertex by substituting the values.
The vertex is (18,66). Hence, the maximum h-value is about 66, meaning that only 66 % of the population will have high-speed Internet. As a result, this quadratic equation is not a good model.
