Write a quadratic function in standard form that passes through the points (-8, 0), (-5, -3), and (-2, 0).
Understand the Problem
The question asks for a quadratic function in standard form that passes through three given points: (-8, 0), (-5, -3), and (-2, 0). We need to find the coefficients of the quadratic function f(x) = ax^2 + bx + c such that the function satisfies all three points.
Answer
$f(x) = \frac{1}{3}x^2 + \frac{10}{3}x + \frac{16}{3}$
Answer for screen readers
$f(x) = \frac{1}{3}x^2 + \frac{10}{3}x + \frac{16}{3}$
Steps to Solve
- Use the roots to write the quadratic in factored form
Since the quadratic passes through $(-8, 0)$ and $(-2, 0)$, we know that $x = -8$ and $x = -2$ are the roots. Thus, the quadratic can be written in factored form as $f(x) = a(x + 8)(x + 2)$ for some constant $a$.
- Solve for $a$ using the point $(-5, -3)$
We are given that the quadratic passes through the point $(-5, -3)$. Substitute $x = -5$ and $f(x) = -3$ into the equation $f(x) = a(x + 8)(x + 2)$: $$ -3 = a(-5 + 8)(-5 + 2) $$ $$ -3 = a(3)(-3) $$ $$ -3 = -9a $$ $$ a = \frac{-3}{-9} = \frac{1}{3} $$ Thus we have $f(x) = \frac{1}{3}(x + 8)(x + 2)$.
- Expand the quadratic to standard form
Now we expand the quadratic to get it into standard form $f(x) = ax^2 + bx + c$: $$ f(x) = \frac{1}{3}(x^2 + 2x + 8x + 16) $$ $$ f(x) = \frac{1}{3}(x^2 + 10x + 16) $$ $$ f(x) = \frac{1}{3}x^2 + \frac{10}{3}x + \frac{16}{3} $$ Therefore, the quadratic function in standard form is $f(x) = \frac{1}{3}x^2 + \frac{10}{3}x + \frac{16}{3}$.
$f(x) = \frac{1}{3}x^2 + \frac{10}{3}x + \frac{16}{3}$
More Information
The problem asked us to identify the quadratic equation, given the points (-8,0), (-5, -3), and (-2,0). The key was to write the quadratic in factored form using the $x$ intercepts and then solve for the leading coefficient using the third point. Expanding into standard form was the last step to write in the $f(x) = ax^2 + bx + c$ form.
Tips
A common mistake is to forget to solve for $a$ after writing the quadratic in factored form. Another common mistake is to make errors when expanding and simplifying the quadratic expression. Carefully distribute and combine like terms to avoid these mistakes.
AI-generated content may contain errors. Please verify critical information
