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Chapter 3: Problem 9

Find the \(x\)- and \(y\)-intercepts of the rational function. $$r(x)=\frac{x^{2}-9}{x^{2}}$$

Short Answer

Expert verified
The x-intercepts are \((3, 0)\) and \((-3, 0)\). There is no y-intercept.

Step by step solution

01

Find the x-intercepts

To find the x-intercepts, set the numerator equal to zero because the x-intercept occurs where the function value is zero. For \( r(x) = \frac{x^2 - 9}{x^2} \), solve the equation \( x^2 - 9 = 0 \).
02

Solve for x-intercepts

Solve the equation \( x^2 - 9 = 0 \). This is a difference of squares, which can be factored as \((x - 3)(x + 3) = 0 \). Set each factor equal to zero: \( x - 3 = 0 \) or \( x + 3 = 0 \). So, \( x = 3 \) or \( x = -3 \). Thus, the x-intercepts are \( (3, 0) \) and \( (-3, 0) \).
03

Find the y-intercept

To find the y-intercept, set \( x = 0 \) and evaluate \( r(x) \). Calculate \( r(0) = \frac{0^2 - 9}{0^2} \). Since the denominator becomes zero, the function is undefined at \( x = 0 \), meaning there is no y-intercept.

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Key Concepts

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

Understanding X-Intercepts
Finding the x-intercepts of a rational function involves identifying the points where the function crosses the x-axis. This happens when the output of the function, or the y-value, is zero. To determine these points, set the numerator of the rational function equal to zero and solve for the variable.
  • For example, consider the function \( r(x) = \frac{x^2 - 9}{x^2} \). To find the x-intercepts, set \( x^2 - 9 = 0 \), since this is the part of the function that can make \( r(x) \) equal to zero.
  • Solving \( x^2 - 9 = 0 \) involves recognizing this as a "difference of squares" which is a special way to factor polynomials.
  • The x-intercepts for this function are at points where the graph touches the x-axis, specifically at \( (3, 0) \) and \( (-3, 0) \).
Recognizing the x-intercepts is crucial as they help shape the graph of the function and reveal key insights into where the graph crosses the axis.
Exploring Y-Intercepts
The y-intercept of a function occurs where the graph crosses the y-axis. At this point, the value of \( x \) is 0. To find the y-intercept of a rational function, substitute 0 for \( x \) in the function and solve for the resulting value.
  • In the function \( r(x) = \frac{x^2 - 9}{x^2} \), attempting to substitute \( x = 0 \) leads to \( r(0) = \frac{0^2 - 9}{0^2} \).
  • This calculation reveals an undefined operation, as the denominator becomes zero, leading to a restriction on the graph where it cannot be evaluated.
  • As a result, this function does not have a y-intercept, since it is undefined at \( x = 0 \).
Understanding y-intercepts involves recognizing where the function cannot exist as well as where it does. This helps to understand the graphical representation of functions better and understand exclusion zones like vertical asymptotes.
The Difference of Squares
The difference of squares is a powerful algebraic concept that simplifies solving polynomial equations. It's characterized by two terms, each a perfect square, separated by a subtraction sign. Identifying and leveraging this structure can make solving certain equations far more straightforward.
  • A typical form of the difference of squares is \( a^2 - b^2 \), which factors into \( (a - b)(a + b) \).
  • In the equation \( x^2 - 9 = 0 \) from our original problem, it readily fits this form since \( x^2 \) and 9 are perfect squares.
  • Breaking it down, \( x^2 - 3^2 = (x - 3)(x + 3) = 0 \), shows how each part can be set to zero to solve for x-intercepts.
Mastering this technique provides an efficient way to handle quadratic equations and is particularly helpful when dealing with rational functions to determine intersections and analyze their behavior.

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Most popular questions from this chapter

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{x^{2}-2 x+1}{x^{2}+2 x+1}$$

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=2 x^{3}-x^{2}+4 x-7$$

The Depressed Cubic The most general cubic (third-degree) equation with rational coefficients can be written as $$x^{3}+a x^{2}+b x+c=0$$ (a) Show that if we replace \(x\) by \(X-a / 3\) and simplify, we end up with an equation that doesn't have an \(X^{2}\) term, that is, an equation of the form $$X^{3}+p X+q=0$$ This is called a depressed cubic, because we have "depressed" the quadratic term. (b) Use the procedure described in part (a) to depress the equation \(x^{3}+6 x^{2}+9 x+4=0\)

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1$$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2$$
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