Problem 3 - Beginning and Intermediate Algebra

Question

How can we determine the number of \(x\) -intercepts of the graph of a quadratic function without graphing the function?

Step-by-Step Solution

Answer

Use the discriminant \(b^2 - 4ac\). If it's >0, there are 2 x-intercepts; if it's 0, there is 1 x-intercept; if it's <0, there are no x-intercepts.

Step 1: Understand the quadratic function

A quadratic function is generally written as: \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The x-intercepts are the points where the graph intersects the x-axis.

Step 2: Set the function equal to zero

To find the x-intercepts, set the quadratic function equal to zero: \(ax^2 + bx + c = 0\). This equation is called the quadratic equation.

Step 3: Use the quadratic formula

The quadratic formula \(x = \frac{-b \, ± \, \sqrt{b^2 - 4ac}}{2a}\) is used to find the roots (x-intercepts) of the quadratic equation.

Step 4: Determine the discriminant

Identify the discriminant \(b^2 - 4ac\) within the quadratic formula. The value of the discriminant indicates the number of x-intercepts: - If \(b^2 - 4ac > 0\), there are two distinct real x-intercepts.- If \(b^2 - 4ac = 0\), there is exactly one real x-intercept (the vertex touches the x-axis).- If \(b^2 - 4ac < 0\), there are no real x-intercepts (the graph does not touch the x-axis).

Key Concepts

x-intercepts

In a quadratic function, the x-intercepts are the points where the graph intersects the x-axis.
These are the solutions to the equation when the function equals zero.
You can imagine x-intercepts as the 'roots' or 'zeros' of the quadratic equation.
To find x-intercepts, we first set the quadratic function, which is usually in the form of:
\( f(x) = ax^2 + bx + c \) equal to zero: \( ax^2 + bx + c = 0 \). These x-values where the function equals zero are your x-intercepts.

quadratic equation

The quadratic equation is a second-degree polynomial equation in a single variable x.
Its standard form is written as: \( ax^2 + bx + c = 0 \). Here, 'a', 'b', and 'c' are constants, with 'a' ≠ 0.
Solving the quadratic equation helps us find the roots or x-intercepts.
One common and powerful method to solve it is using the quadratic formula: \( x = \frac{-b \, ± \, \sqrt{b^2 - 4ac}}{2a} \).
The solutions to this formula are where the quadratic graph touches or crosses the x-axis.

discriminant

The discriminant is a component of the quadratic formula and is given by the expression \( b^2 - 4ac \).
The discriminant helps determine the nature and number of the roots of the quadratic equation.
By examining the discriminant, we can understand whether the quadratic function has:

• Two distinct real x-intercepts (if \( b^2 - 4ac > 0 \) )
• One real x-intercept (if \( b^2 - 4ac = 0 \) )
< li>No real x-intercepts (if \( b^2 - 4ac < 0 \) )

The discriminant tells us a lot with only one calculation, providing essential insight even before we graph the function.