Problem 74 - Algebra 2

Question

How many times does the graph of \(y=x^{2}-4 x+5\) cross the \(x\) -axis? $$ \begin{array}{lllll}{\text { F. } 0} & {\text { 6. } 1} & {\text { H. } 2} & {\text { 1. } 33}\end{array} $$

Step-by-Step Solution

Answer

The graph of the function does not cross the x-axis.

Step 1: Understanding the x-axis intersection

The graph of a function crosses the x-axis at points where the y value is zero. This means we need to find the roots of the equation by setting it equal to zero.

Step 2: Setting the function equal to zero

Set the function equal to zero to find its roots: \(x^{2}-4x+5=0\).

Step 3: Determining the number of real roots

Use the discriminant \(b^{2}-4ac\) from the quadratic formula \(\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) to determine the number of real roots.

Step 4: Calculating the discriminant

Calculate the discriminant for the given quadratic equation: \(b^{2}-4ac=(-4)^{2}-4(1)(5)=16-20=-4\).

Step 5: Interpreting the discriminant

A negative discriminant indicates that the quadratic equation has no real roots, hence the graph does not cross the x-axis.

Key Concepts

X-axis Intersection

In algebra, the graph of a quadratic equation such as y=ax^2+bx+c often comes up as a parabola. An important feature of this parabola is where it might cross the x-axis. These intersections are points where the graph's height, y, equals zero. Such points are significant because they are the real roots or solutions of the equation.

When you're given a quadratic equation like y=x^2-4x+5, to find the x-axis intersections, you'll need to set y to zero and solve x^2-4x+5=0. The roots of this equation, if they exist, will tell you the exact point(s) where the parabola touches the x-axis. If a quadratic equation has no real roots, it means the graph doesn't intersect the x-axis at all, floating above or below it instead.

Quadratic Formula

Solving for the roots of a quadratic equation can be done most effectively using the quadratic formula: \(\frac{-b\text{±}\text{sqrt}{(b^2-4ac))}}{2a}\). This powerful tool allows you to find the roots of any quadratic equation ax^2+bx+c=0 by simply substituting the coefficients a, b, and c into the formula. It's particularly useful when the equation isn't easily factored.

The ± sign in the formula denotes that there can be two solutions: one where you add the square root term, and another where you subtract it. When using this formula, pay special attention to the discriminant—\(b^{2}-4ac\)—placed under the square root. This discriminant is key in determining the nature and number of the roots.

Real Roots

Real roots are the solutions of a quadratic equation that are real numbers. The discriminant (\(b^2-4ac\)) of the quadratic formula plays a decisive role in determining the nature of the roots. Its value indicates:

• Discriminant > 0: Two distinct real roots.
• Discriminant = 0: One real root, also known as a repeated or double root.
• Discriminant < 0: No real roots; instead, the roots are complex numbers.

For the equation x^2-4x+5=0, the discriminant is calculated as \((-4)^2-4(1)(5)\), which simplifies to \(-4\). Since this is negative, it tells us there are no real roots, and therefore, the graph does not intersect the x-axis at any point. This is crucial information because it informs us about the parabola's position relative to the x-axis – in this case, it does not touch it at all.