Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=(x-4)(x+2)\)

Step-by-Step Solution

Answer

The x-intercepts are 4 and -2, and the y-intercept is -8.

Step 1: Finding the x-intercepts

The x-intercepts are found by setting the equation equal to zero. That is, to solve the equation \(y = (x-4)(x+2)= 0\), set \(x - 4 = 0\) and \(x + 2 = 0\), then solve for \(x\). This gives \(x = 4\) and \(x = -2\). So the x-intercepts are 4 and -2.

Step 2: Finding the y-intercept

The y-intercept is found by setting \(x = 0\) in the equation and solving for \(y\). That is, plug \(x = 0\) into the equation \(y = (x - 4)(x + 2)\) to find \(y\). This gives \(y = (0 - 4)(0 + 2) = -8\). So the y-intercept is -8.

Key Concepts

Solving Quadratic Equations

Understanding the method of solving quadratic equations is crucial for many aspects of algebra, including finding the x-intercepts of a function. A quadratic equation is typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants.

There are multiple ways to solve a quadratic equation: factoring, using the quadratic formula, completing the square, or graphing. In the context of our exercise, the equation is factored into y = (x - 4)(x + 2). The solutions for x when y equals zero are the x-intercepts of the graph, which are important points where the graph crosses the x-axis.

Remember, factoring is often the quickest method when the quadratic can be easily decomposed into binomials, as seen in this exercise with x intercepts at 4 and -2. It's like breaking the equation down into simpler pieces to find where the graph touches the x-axis.

Graphing Parabolas

Graphing parabolas is a visual way to comprehend the behavior of quadratic functions. A parabola is the graph formed from a quadratic equation and is a U-shaped curve. When graphing y = (x - 4)(x + 2), you'll see its shape and important features, like the vertex, axis of symmetry, and intercepts.

When you graph this particular quadratic equation, you'll plot the x-intercepts at 4 and -2 where the curve crosses the x-axis. The y-intercept at -8 is where the curve crosses the y-axis. These points are crucial for sketching the parabola accurately. To graph a parabola:

• Find the vertex, the highest or lowest point on the graph, which is mid-way between the x-intercepts and can be calculated using the formula x = -b / (2a) for a parabola in standard form.
• Draw the axis of symmetry, a vertical line that runs through the vertex and divides the parabola into two mirror images.
• Plot the intercepts and vertex on a coordinate grid.
• Sketch the parabola using the intercepts and vertex as guide points.

Finding Intercepts of a Function

Finding the intercepts of a function is a fundamental skill in graphing, as they provide key points that help you understand and draw the graph. The x-intercepts, or roots, occur where the graph crosses the x-axis. The y-intercept is the point where the graph crosses the y-axis.

To find the x-intercepts, as demonstrated in the exercise, set y equal to zero and solve for x. Conversely, to find the y-intercept, set x equal to zero and solve for y, which yields a single value since a function can have at most one y-intercept. These intercepts give you an initial sense of the function's graph, especially for linear and quadratic functions.

Intercepts are not just points but represent the function's behavior. For example, in the equation y = (x - 4)(x + 2), the x-intercepts 4 and -2 tell us the function changes sign, and the y-intercept -8 gives the starting point for graphing. Recognizing these will enrich your understanding of how the graph relates to its algebraic expression.