How to find the vertex form of a quadratic function?
Understand the Problem
The question is asking for the method to convert a quadratic function into its vertex form. This typically involves using the process of completing the square or applying the vertex formula to identify the vertex coordinates and rewrite the function accordingly.
Answer
The vertex form is $f(x) = a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a} \right)$ with vertex $(h, k)$ given by $h = -\frac{b}{2a}, k = c - \frac{b^2}{4a}$.
Answer for screen readers
The vertex form of the quadratic function $f(x) = ax^2 + bx + c$ is
$$ f(x) = a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a} \right) $$
with vertex coordinates:
$$ h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a} $$
Steps to Solve
- Identify the standard quadratic form
Start with the standard form of the quadratic function, which is given as
$$ f(x) = ax^2 + bx + c $$
where $a$, $b$, and $c$ are constants.
- Complete the square
To convert it to vertex form, we will complete the square. We will first factor out $a$ from the first two terms of the equation:
$$ f(x) = a \left( x^2 + \frac{b}{a} x \right) + c $$
- Find the term to complete the square
Next, to complete the square inside the parentheses, we need to add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is $\frac{b}{a}$, so half of it is $\frac{b}{2a}$. Therefore, we will add and subtract $(\frac{b}{2a})^2$:
$$ f(x) = a \left( x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 \right) + c $$
This simplifies to:
$$ f(x) = a \left( \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 \right) + c $$
- Simplify the equation
Now, distribute $a$ and combine the constant terms:
$$ f(x) = a \left( x + \frac{b}{2a} \right)^2 - a \left( \frac{b}{2a} \right)^2 + c $$
This gives us:
$$ f(x) = a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a} \right) $$
- Write in vertex form
Now, we can write it in vertex form, which is:
$$ f(x) = a (x - h)^2 + k $$
where the vertex $(h, k)$ is given by:
$$ h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a} $$
Thus, the vertex form is:
$$ f(x) = a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a} \right) $$
The vertex form of the quadratic function $f(x) = ax^2 + bx + c$ is
$$ f(x) = a \left( x + \frac{b}{2a} \right)^2 + \left( c - \frac{b^2}{4a} \right) $$
with vertex coordinates:
$$ h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a} $$
More Information
Vertex form of a quadratic function allows you to easily identify the vertex of the parabola, which is useful in graphing and understanding the function's properties. The vertex represents the maximum or minimum point of the parabola, depending on the value of $a$.
Tips
- Forgetting to factor out $a$: If $a \neq 1$, be sure to factor it out before completing the square to avoid errors.
- Neglecting to adjust constants: When completing the square, it's important to add and subtract the same value properly.
- Incorrect signs for the vertex coordinates: Remember that $h$ is negative of $\frac{b}{2a}$, so watch for sign errors.
AI-generated content may contain errors. Please verify critical information
