Problem 29 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$f(x)=x^{2}-9$$

Step-by-Step Solution

Answer

The vertex is at (0, -9).

Step 1 - Identify the quadratic function

The given quadratic function is \[f(x) = x^2 - 9.\]

Step 2 - Write the quadratic function in standard form

The standard form of a quadratic function is \[f(x) = ax^2 + bx + c.\] For the given function, we see that \[a=1,\ b=0,\ and\ c=-9.\]

Step 3 - Using the vertex formula

The x-coordinate of the vertex of a quadratic function in standard form can be found using the formula \[x = -\frac{b}{2a}.\] In this case, since \(b=0\) and \(a=1\), the formula simplifies to \[x = -\frac{0}{2(1)} = 0.\]

Step 4 - Find the y-coordinate by plugging the x-value back into the function

Now substitute \(x=0\) back into the function: \[f(0) = (0)^2 - 9 = -9.\] Therefore, the y-coordinate is -9.

Step 5 - Write the vertex

The vertex of the quadratic function is at the point (0, -9).

Key Concepts

quadratic function

A quadratic function is a type of polynomial function with the highest degree of 2. This means its highest exponent is 2.
The general form of a quadratic function is \[f(x) = ax^2 + bx + c\], where

• a, b, and c are constants
• x represents the variable

Quadratic functions are characterized by their U-shaped graphs called parabolas.
The direction of the parabola depends on the value of 'a'. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.

standard form

The standard form of a quadratic function is useful for identifying key features such as the vertex and the axis of symmetry.
The standard form is given by: \[f(x) = ax^2 + bx + c\].
To identify a quadratic function in standard form, locate the

• Coefficient 'a' in front of \(x^2\)
• Coefficient 'b' in front of x
• Constant term 'c'

In our given function \(f(x) = x^2 - 9\), the coefficients are

• a = 1
• b = 0
• c = -9

vertex formula

The vertex of a quadratic function is the point where the parabola either reaches its maximum or minimum value.
To find the x-coordinate of the vertex, we use the vertex formula: \[x = -\frac{b}{2a}\].
For the function \(f(x) = x^2 - 9\), where \(a = 1\) and \(b = 0\):

• Plug in the values into the formula: \[x = -\frac{0}{2(1)} = 0\]

The x-coordinate of the vertex is 0. Next, substitute \(x = 0\) back into the original function to find the y-coordinate.

function evaluation

Function evaluation is the process of finding the output of a function for a specific input value.
To complete this step for our problem, we substitute \(x = 0\) into the function \(f(x) = x^2 - 9\): \[f(0) = (0)^2 - 9 = -9\].
Here, the output or y-coordinate is -9.
Therefore, the vertex of the quadratic function \(f(x) = x^2 - 9\) is at the point (0, -9), which tells us where the lowest point of the parabola is.