How to find vertex from factored form?
Understand the Problem
The question is asking for the method to determine the vertex of a quadratic function in its factored form, which typically involves identifying the roots and calculating the midpoint to find the vertex's x-coordinate, then evaluating the function to get the corresponding y-coordinate.
Answer
The vertex is $\left( \frac{r_1 + r_2}{2}, f\left(\frac{r_1 + r_2}{2}\right) \right)$.
Answer for screen readers
The vertex of the quadratic function in factored form is given as the ordered pair $\left( x_{\text{vertex}}, y_{\text{vertex}} \right)$, where:
$$ x_{\text{vertex}} = \frac{r_1 + r_2}{2} $$
and
$$ y_{\text{vertex}} = f\left(\frac{r_1 + r_2}{2}\right) $$.
Steps to Solve
- Identify the roots from the factored form
Given a quadratic function in factored form, it typically looks like $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots. Identify the values of $r_1$ and $r_2$.
- Calculate the x-coordinate of the vertex
The x-coordinate of the vertex can be found by taking the average of the roots. Use the formula:
$$ x_{\text{vertex}} = \frac{r_1 + r_2}{2} $$
- Evaluate the function to find the y-coordinate
Substitute the x-coordinate of the vertex back into the original quadratic function to find the corresponding y-coordinate:
$$ y_{\text{vertex}} = f\left(\frac{r_1 + r_2}{2}\right) $$
- Write the vertex as an ordered pair
Once you have both the x- and y-coordinates, the vertex can be represented as an ordered pair $\left( x_{\text{vertex}}, y_{\text{vertex}} \right)$.
The vertex of the quadratic function in factored form is given as the ordered pair $\left( x_{\text{vertex}}, y_{\text{vertex}} \right)$, where:
$$ x_{\text{vertex}} = \frac{r_1 + r_2}{2} $$
and
$$ y_{\text{vertex}} = f\left(\frac{r_1 + r_2}{2}\right) $$.
More Information
The vertex of a quadratic function represents the highest or lowest point on the graph, depending on the direction of the parabola. If the coefficient $a$ is positive, the parabola opens upwards, and the vertex is a minimum point. Conversely, if $a$ is negative, the parabola opens downwards and the vertex is a maximum point.
Tips
- Confusing the order of roots: Make sure $r_1$ and $r_2$ are treated correctly when calculating the average.
- Forgetting to substitute the x-coordinate back into the function: Always evaluate the function at the x-coordinate to get the y-coordinate of the vertex.
