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Chapter 2: Problem 89

Draw a graph of an equation that contains two \(x\) -intercepts; at one the graph crosses the \(x\) -axis, and at the other the graph touches the \(x\) -axis.

Short Answer

Expert verified
Use y = (x-2)(x+1)^2 for the graph. It crosses the x-axis at x=2 and touches at x=-1.

Step by step solution

01

Identify the Type of Polynomial Needed

To create a graph with two different behaviors at the x-intercepts, we need a polynomial. A quadratic polynomial can meet our requirements if we allow for a repeated root.
02

Choose a Quadratic Polynomial

A sample quadratic polynomial with one crossing intercept and one touching intercept is: y = (x - a)(x - b)^2 where 'a' is the intercept where the graph crosses the x-axis and 'b' is the intercept where the graph touches the x-axis.
03

Assign Specific Values

Choose specific values for 'a' and 'b' to get concrete points. For instance, let 'a' be 2 and 'b' be -1. Thus, the equation becomes: y = (x - 2)(x + 1)^2
04

Draw the Graph

Plot the intercepts on the x-axis at x = 2 and x = -1. Draw the curve so that it crosses the x-axis at x = 2 and touches the x-axis at x = -1. Make sure to show that near x = -1, the curve just touches the x-axis without crossing it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercepts
In the context of graphing polynomial functions, x-intercepts are the points where the polynomial graph crosses the x-axis. These points occur when the value of y is zero. For a polynomial function, the x-intercepts are the solutions to the equation when set to zero. For example, in the polynomial equation y = (x - 2)(x + 1)^2, the x-intercepts are the points where the polynomial equals zero: x = 2 and x = -1. The x-intercept at x = 2 is called a 'crossing intercept,' and the one at x = -1 is called a 'touching intercept,' due to the different behaviors of the graph at these points.
crossing intercept
A crossing intercept is an x-intercept where the graph actually crosses the x-axis. This means the polynomial changes sign at this point, moving from positive to negative or vice versa. For instance, in the given polynomial y = (x - 2)(x + 1)^2, the point x = 2 is a crossing intercept. To spot a crossing intercept, look for where the graph changes direction from above to below the x-axis or from below to above.
touching intercept
A touching intercept, also known as a 'tangent intercept', is an x-intercept where the graph touches the x-axis but does not cross it. Instead of changing sign, the function reaches zero, touches the x-axis momentarily, and then turns back to its original direction. In the polynomial y = (x - 2)(x + 1)^2, the point x = -1 is a touching intercept. This behavior typically indicates a repeated root in the polynomial, meaning the root has an even multiplicity. This makes the graph flatten out at the intercept before turning back.
graphing polynomial functions
Graphing polynomial functions involves plotting points calculated from the polynomial equation to visualize its shape. Polynomials of different degrees have characteristic shapes. Quadratic polynomials, for example, are represented by parabolas. A polynomial in the form y = (x - a)(x - b)^2 will have distinct x-intercepts with varying behaviors. When graphing, ensure to mark and label the x-intercepts clearly, noting whether they are crossing or touching. From our example, we see a crossing intercept at x = 2, where the graph moves through the x-axis, and a touching intercept at x = -1, where the graph touches the x-axis and turns back.

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