Question

The \(x\) -coordinate of the vertex of \(f(x)=a x^{2}+b x+c, a \neq 0,\) is __________.

Short answer

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

Step-by-step solution

- Identify Vertex Formula

The vertex of a quadratic function given by the equation \( f(x) = ax^2 + bx + c \) can be found using the vertex formula. The formula for the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \).

- Apply the Formula

Given the quadratic function \( f(x) = ax^2 + bx + c \), identify the coefficients \( a \) and \( b \). Substituting these into the formula \( x = -\frac{b}{2a} \) gives the x-coordinate of the vertex.

- Simplify

Simplify the expression \( x = -\frac{b}{2a} \) to find the x-coordinate. This value is the x-coordinate of the vertex for the given quadratic function.

Key concepts

vertex formula

When dealing with quadratic functions, finding the vertex is key to understanding the graph's highest or lowest point.
The vertex formula specifically helps us determine the x-coordinate of this point. A quadratic function generally looks like this: \( f(x) = ax^2 + bx + c \).
The formula we use to find the x-coordinate of the vertex is \( x = - \frac {b}{2a} \). Here, \(a\) and \(b\) are the coefficients from your quadratic function. This powerful formula provides a direct route to determining the x-coordinate of the vertex in one simple step.
Knowing this coordinate helps you draw the correct shape and position of your quadratic graph.

quadratic function

A quadratic function is a polynomial of degree 2. It has the standard form:\( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants and \(a e 0\).
Let’s break down each part:

• \(ax^2\) : The squared term makes the graph a parabola. The sign of \(a\) determines if it opens upward (positive \(a\)) or downward (negative \(a\)).
  
• \(bx\) : The linear term affects the slope and the axis of symmetry of the parabola.
  
• \(c\) : The constant term represents the y-intercept, where the graph crosses the y-axis.
  

Understanding these elements helps you interpret the graph's shape and position.

x-coordinate of the vertex

The x-coordinate of the vertex plays a crucial role in graphing quadratic functions. Using the vertex formula \( x = - \frac {b}{2a} \), you can pinpoint this important value.
Let’s go through an example:
Suppose we have a quadratic function \( f(x) = 2x^2 + 4x + 1 \). In this case, \(a = 2\) and \(b = 4\). Substituting these into the formula gives:
\( x = - \frac {4}{2 \cdot 2} = - \frac{4}{4} = -1 \).
This value, -1, is the x-coordinate of the vertex. It tells us where the axis of symmetry for the parabola is located on the x-axis.

simplification

Simplification is important when working with formulas to ensure clear, easy-to-use results.
After substituting the values of \(a\) and \(b\) into the vertex formula, you may need to simplify the expression.
Here's a quick refresher on the simplification process:

• Apply any arithmetic operations (like multiplication or division) inside the fraction.
  
• Reduce the fraction to its simplest form. This step makes it easier to understand and use the value.
  

For example, in the previous case with \( f(x) = 2x^2 + 4x + 1 \), after substitution, we had:\( - \frac{4}{2 \cdot 2} = - \frac {4}{4} = -1 \).
Simplifying the division provided a straightforward answer, giving us a clear and usable x-coordinate for the vertex.