Problem 39 - Algebra 1: Concepts and Skills

Question

Determine whether the graph of the function will intersect the x-axis in zero, one, or two points. \(y=x^{2}-2 x+4\)

Step-by-Step Solution

Answer

Since the discriminant is less than zero, the graph of the function \(y=x^{2}-2 x+4\) does not intersect the x-axis. Therefore, there are no points where it intersects the x-axis.

Step 1: Writing the function and setting it equal to zero

The given function is \(y=x^{2}-2 x+4\). To find the x-intercepts, set \(y=0\). So we get the equation \(x^{2}-2 x+4=0\)

Step 2: Identifying the coefficients

Here the coefficients are a=1, b=-2, and c=4 according to the general quadratic function \(ax^{2}+bx+c\). These are going to be plugged into the discriminant formula.

Step 3: Calculating the discriminant

The formula for the discriminant is \[b^{2}-4ac\]. Substituting a=1, b=-2, and c=4 into the formula we get \( (-2)^{2}-4\cdot1\cdot4 = 4 - 16 = -12 \]

Step 4: Analysis of the discriminant

As the discriminant is -12, which is less than 0, it indicates that there are no real solutions to the equation \(x^{2}-2 x+4=0\).

Key Concepts

Understanding X-Intercepts

To find where a parabola intersects the x-axis, we look for x-intercepts. These occur where the graph of the function crosses the x-axis. Typically, x-intercepts happen when the output, or y-value, is zero. So, if we're given a quadratic function like this one:

• \( y = x^2 - 2x + 4 \)

we can set \( y = 0 \) to find the x-intercepts:

• \( x^2 - 2x + 4 = 0 \)

This equation helps us identify potential points where the graph touches or crosses the x-axis. By solving the quadratic equation, you can find how many and what type of x-intercepts exist. In some scenarios, there might be no real x-intercepts at all, indicating the graph does not touch or cross the x-axis.

Evaluating the Discriminant

A pivotal part of dealing with quadratic equations is understanding the discriminant. This is a component of the quadratic formula used to determine how many real solutions the equation has. The discriminant is derived from the formula:

• \( b^2 - 4ac \)

For the quadratic equation \( x^2 - 2x + 4 \), the coefficients are:

• \( a = 1 \)
• \( b = -2 \)
• \( c = 4 \)

Substituting these into the discriminant formula gives us:

• \( (-2)^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12 \)

When the discriminant is negative, as it is here, it means there are no real solutions. This negative result indicates our quadratic equation does not intersect the x-axis at any real point, visualizing why the graph doesn't touch the x-axis.

Real Solutions in Quadratics

Real solutions refer to the actual x-values that satisfy the quadratic equation when the quadratic expression equals zero. Finding real solutions often involves the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In simpler terms, these are the x-values at which the graph of a quadratic function intersects the x-axis. However, the discriminant, \( b^2 - 4ac \), dictates how many real solutions exist:

• Positive Discriminant: Two real solutions, meaning two x-intercepts.
• Zero Discriminant: One real solution, meaning the graph touches the x-axis at one point and represents a double root.
• Negative Discriminant: No real solutions, indicating the graph does not intersect with the x-axis.

In our example of \( y = x^2 - 2x + 4 \), with a discriminant of -12, there are no real solutions. Understanding the discriminant helps to predict the number of times the function graph will cross the x-axis.