Problem 12 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ y=x^{2}-5 x+6 $$

Step-by-Step Solution

Answer

The x-intercepts are (2, 0) and (3, 0); the y-intercept is (0, 6).

Step 1: Understanding the Problem

We need to find the x- and y-intercepts of the given quadratic equation: \( y = x^2 - 5x + 6 \). The x-intercepts are the points where the graph crosses the x-axis (\( y = 0 \)), and the y-intercept is where the graph crosses the y-axis (\( x = 0 \)).

Step 2: Finding the X-Intercepts

To find the x-intercepts, we set \( y = 0 \) in the equation \( y = x^2 - 5x + 6 \) and solve for \( x \). This leads to the quadratic equation \( x^2 - 5x + 6 = 0 \).

Step 3: Solving the Quadratic Equation

We can factor the quadratic equation \( x^2 - 5x + 6 = 0 \) into \((x - 2)(x - 3) = 0\). Solving each factor for zero gives us \( x - 2 = 0 \) and \( x - 3 = 0 \).

Step 4: Calculating the X-Intercepts

Solving the equations \( x - 2 = 0 \) and \( x - 3 = 0 \) gives us \( x = 2 \) and \( x = 3 \). Therefore, the x-intercepts of the graph are the points \((2, 0)\) and \((3, 0)\).

Step 5: Finding the Y-Intercept

To find the y-intercept, we set \( x = 0 \) in the original equation \( y = x^2 - 5x + 6 \). This gives us \( y = 0^2 - 5(0) + 6 = 6 \).

Step 6: Concluding the Y-Intercept

When \( x = 0 \), we found \( y = 6 \). Therefore, the y-intercept of the graph is the point \((0, 6)\).

Key Concepts

x-intercepts

The x-intercepts of a quadratic equation are the points where the graph crosses the x-axis. At these points, the value of the equation is zero because

• the graph touches or cuts the x-axis
• the value of y is zero along the x-axis

To find these intersections, we set the equation to zero and solve for x. For example, if we have the quadratic equation \( y = x^2 - 5x + 6 \), we replace \( y \) with 0: \[ 0 = x^2 - 5x + 6 \] This forms a new equation which we need to solve to find the values of \( x \) where this condition holds true. These values are the x-intercepts of the equation.

y-intercepts

The y-intercept is the point where the graph crosses the y-axis. Unlike the x-intercepts, here the value of x is always 0 because:

• the graph meets the y-axis
• the value of y can change depending on the equation

To find the y-intercept, simply substitute \( x = 0 \) into the equation. Using our example quadratic equation \( y = x^2 - 5x + 6 \), we set \( x \) to 0: \[ y = 0^2 - 5(0) + 6 = 6 \] Thus, the y-intercept of this equation is the point \((0, 6)\). It represents the starting value of the quadratic function on the y-axis.

factoring quadratic equations

Factoring quadratic equations is a method used to simplify and solve them. We look for two numbers that both add up to the coefficient of the middle term and multiply to the constant term. This is the sum-product method.

• Write the equation in the standard form \(ax^2 + bx + c = 0\)
• Identify a pair of factors of the product \(ac\) that add up to \(b\)

For the equation \(x^2 - 5x + 6 = 0\), we need two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of \(x\)). These numbers are -2 and -3. This means we can factor the equation as: \[ (x - 2)(x - 3) = 0 \] Now, solving the quadratic equation becomes a matter of solving each factor set to zero.

solving quadratic equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. Factoring is often the first method used as it is simple and effective when applicable. Once a quadratic equation is factored, it can be set up as a product equal to zero:

• Use the Zero Product Property
• Set each factor to zero and solve them

Using our factorized equation \((x - 2)(x - 3) = 0\), we apply the Zero Product Property. This property states that if a product of factors is zero, at least one of the factors must be zero. So we solve these two simple equations: \[ \begin{align*} x - 2 &= 0 \ x - 3 &= 0 \end{align*} \] Solving these gives us the x-intercepts, \( x = 2 \) and \( x = 3 \), which are the solutions to the quadratic equation.