High School Math : Graphing Parabolic Functions

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Pre Calculus

What is the minimal value of 

\(\displaystyle 2x^2+16x-7\)

over all real numbers?

Possible Answers:

\(\displaystyle -39\)

No minimum value.

\(\displaystyle -10\)

\(\displaystyle 2\)

\(\displaystyle 0\)

Correct answer:

\(\displaystyle -39\)

Explanation:

Since this is an upwards-opening parabola, its minimum value will occur at the vertex.  The \(\displaystyle x\)-coordinate for the vertex of any parabola of the form

\(\displaystyle y = ax^2 + bx+c\)

is at

\(\displaystyle x = \frac{-b}{2a}\)

 

So here, 

\(\displaystyle x = \frac{-b}{2a}\)

\(\displaystyle = \frac{-(16)}{2(2)}\)

\(\displaystyle = -4\)

 

We plug this value back into the equation of the parabola, to find the value of the function at this \(\displaystyle x\)

\(\displaystyle y = 2(-4)^2 + 16(4) - 7\)

\(\displaystyle =-39\)

Thus the minimal value of the expression is \(\displaystyle -39\)

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