In Exercises 83-86, explain the mistake that is made. There...

In Exercises $83-86,$ explain the mistake that is made. There may be a single mistake or there may be more than one mistake. Determine the vertex of the quadratic function $$ f(x)=2 x^{2}-6 x-18 $$ Solution: Step 1: The vertex is given by $(h, k)=\left(-\frac{b}{2 a^{\prime}} f\left(-\frac{b}{2 a}\right)\right)$ In this case, $a=2$ and $b=6$ Step 2: The $x$ -coordinate of the vertex is $$ x=-\frac{6}{2(2)}=-\frac{6}{4}=-\frac{3}{2} $$ Step 3: The $y$ -coordinate of the vertex is $$ \begin{aligned} f\left(-\frac{3}{2}\right) &=-2\left(-\frac{3}{2}\right)^{2}+6\left(-\frac{3}{2}\right)-18 \\ &=-2\left(\frac{9}{4}\right)-\frac{18}{2}-18 \\ &=-\frac{9}{2}-9-18 \\ &=-\frac{63}{2} \end{aligned} $$ This is incorrect. What mistake(s) was made?

In Exercises $83-86,$ explain the mistake that is made. There may be a single mistake or there may be more than one mistake.
Determine the vertex of the quadratic function
$$
f(x)=2 x^{2}-6 x-18
$$
Solution:
Step 1: The vertex is given by $(h, k)=\left(-\frac{b}{2 a^{\prime}} f\left(-\frac{b}{2 a}\right)\right)$ In this case, $a=2$ and $b=6$
Step 2: The $x$ -coordinate of the vertex is
$$
x=-\frac{6}{2(2)}=-\frac{6}{4}=-\frac{3}{2}
$$
Step 3: The $y$ -coordinate of the vertex is
$$
\begin{aligned}
f\left(-\frac{3}{2}\right) &=-2\left(-\frac{3}{2}\right)^{2}+6\left(-\frac{3}{2}\right)-18 \\
&=-2\left(\frac{9}{4}\right)-\frac{18}{2}-18 \\
&=-\frac{9}{2}-9-18 \\
&=-\frac{63}{2}
\end{aligned}
$$
This is incorrect. What mistake(s) was made?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Quadratic Function Standard Form

A quadratic function is typically written in the standard form ax² + bx + c, where a, b, and c are constants. Correctly identifying these coefficients is essential, as they determine the shape and key features of the graph, including the vertex and axis of symmetry.

Coefficient Identification

Correctly extracting the coefficients a, b, and c from the quadratic expression is crucial. In this problem, the mistake was to misidentify b as 6 instead of -6. This error affects subsequent calculations, particularly those that involve the vertex formula.

Vertex Formula

The vertex of a quadratic function can be found using the formula x = -b/(2a), where substituting the correct values of a and b is critical. An error in determining b leads to an incorrect x-coordinate, fundamentally altering the correct location of the vertex.

Function Evaluation at the Vertex

After computing the x-coordinate of the vertex, it is necessary to substitute this value back into the function to determine the y-coordinate. Mistakes in basic arithmetic or sign errors during this substitution, as seen in the solution, result in an inaccurate calculation of the vertex's y-component.

Transcript

Please give Ace some feedback