Certainly! Let’s guide Martha through correcting her homework assignment involving the function [tex]\( f(x) = 5x^3 + 2x^2 + 7x - 3 \)[/tex].
A quadratic function is in the form [tex]\( ax^2 + bx + c \)[/tex], where the highest power of [tex]\(x\)[/tex] is 2. The function she wrote includes a cubic term [tex]\( 5x^3 \)[/tex], which needs to be addressed to convert it into a proper quadratic function.
Here are the steps we need to follow to review the possible changes:
1. Elimination of the cubic term:
One way to convert the given function into a quadratic function is to eliminate the cubic term [tex]\( 5x^3 \)[/tex]. This means removing [tex]\( 5x^3 \)[/tex] entirely from the function:
[tex]\[
f(x) = 2x^2 + 7x - 3
\][/tex]
Now, this fits the form of a quadratic function.
2. Change the exponent of the cubic term and combine it with the quadratic term:
Another way to correct her function is to change the exponent of the cubic term [tex]\( 5x^3 \)[/tex] to 2. By doing this, we turn it into another quadratic term with the same variable exponent as the [tex]\( 2x^2 \)[/tex] term:
[tex]\[
5x^2 + 2x^2 = 7x^2
\][/tex]
Thus, the new function becomes:
[tex]\[
f(x) = 7x^2 + 7x - 3
\][/tex]
Now this is also in the form of a quadratic function.
Therefore, the possible changes Martha can make to correct her homework assignment are:
1. The first term, [tex]\( 5x^3 \)[/tex], can be eliminated.
2. The exponent on the first term, [tex]\( 5x^3 \)[/tex], can be changed to a 2 and then combined with the second term, [tex]\( 2x^2 \)[/tex].
These two options are valid to convert her original function into a quadratic one. The other options mentioned do not result in a quadratic function and therefore are not appropriate corrections.
