Problem 39 - Algebra A Combined Function

Question

Find the vertex of the graph of each function. $$ f(x)=x^{2}+2 $$

Step-by-Step Solution

Answer

The vertex of the function is \((0, 2)\).

Step 1: Identify the Standard Form

The given function \( f(x) = x^2 + 2 \) is a quadratic function. A general quadratic function is written in the standard form \( f(x) = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 0 \), and \( c = 2 \).

Step 2: Use the Vertex Formula

The vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = 0 \) into this formula gives \( x = -\frac{0}{2(1)} = 0 \).

Step 3: Find the Vertex's y-coordinate

Substitute \( x = 0 \) back into the function to find the corresponding \( y \)-coordinate. So, \( f(0) = 0^2 + 2 = 2 \).

Step 4: Write the Vertex

The vertex is the point \( (x, y) \) we found in the previous steps. Therefore, the vertex of the function \( f(x) = x^2 + 2 \) is \( (0, 2) \).

Key Concepts

Quadratic Function

A quadratic function is a special type of polynomial that has the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This function represents a parabola when graphed on a coordinate plane.
The shape of the parabola depends on the value of \( a \).

• If \( a > 0 \), the parabola opens upward.
• If \( a < 0 \), the parabola opens downward.

Parabolas have an important point known as the vertex. The vertex can be either the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards.

Standard Form of a Quadratic

The standard form of a quadratic function is expressed as \( ax^2 + bx + c \). This form is particularly useful because it allows us to easily identify key characteristics of the quadratic function.

• \( a \) determines the direction and "width" of the parabola.
• \( b \) influences the horizontal placement of the vertex.
• \( c \) represents the y-intercept, the point where the graph intersects the y-axis.

In the example \( f(x) = x^2 + 2 \), we have \( a = 1 \), \( b = 0 \), and \( c = 2 \). This indicates a parabola that opens upwards with its vertex somewhere along the y-axis. The simplicity of inputting these values into the vertex formula makes it straightforward to locate the vertex.

Vertex Formula

To find the vertex of a quadratic function when it is in standard form, we utilize the vertex formula. The x-coordinate of the vertex is calculated using the formula \( x = -\frac{b}{2a} \).
For our function \( f(x) = x^2 + 2 \), substituting \( b = 0 \) and \( a = 1 \) gives us \( x = -\frac{0}{2(1)} = 0 \). This tells us that the vertex is located on the y-axis.
Once we have the x-coordinate of the vertex, we substitute it back into the original function to find the corresponding y-coordinate. For \( f(x) = x^2 + 2 \), substituting \( x = 0 \) gives \( f(0) = 0^2 + 2 = 2 \), yielding the vertex \( (0, 2) \).
Understanding this process allows us to quickly and efficiently determine the vertex of any parabolic function presented in standard form.