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Chapter 3: Problem 7

Find the \(x\)- and \(y\)-intercepts of the rational function. $$t(x)=\frac{x^{2}-x-2}{x-6}$$

Short Answer

Expert verified
The x-intercepts are (2,0) and (-1,0); the y-intercept is (0, 1/3).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the numerator of the function equal to zero, since the x-intercept occurs where the function crosses the x-axis ( t(x) = 0). Here, we solve for x in the equation:\[ x^2 - x - 2 = 0 \]We can factor this equation to find its roots:\[ (x - 2)(x + 1) = 0 \]Set each factor equal to zero:\[ x - 2 = 0 \quad \text{or} \quad x + 1 = 0 \]Thus, \( x = 2 \) or \( x = -1 \). These are the x-intercepts: (2,0) and (-1,0).
02

Find the y-intercept

To find the y-intercept, set x equal to zero in the original function and solve for t(x), since the y-intercept occurs where the function crosses the y-axis. Substitute:\[ t(0) = \frac{0^2 - 0 - 2}{0 - 6} = \frac{-2}{-6} = \frac{1}{3} \]The y-intercept is at (0, 1/3).

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Key Concepts

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

Understanding X-Intercept
The x-intercept of a function is the point where the graph of the function crosses the x-axis. For rational functions like \( t(x)=\frac{x^{2}-x-2}{x-6} \), the x-intercept can be found by setting the numerator equal to zero. This is because at the x-intercept, the value of the function is zero, meaning:
  • The function equals zero at the x-intercept.
  • The graph crosses the x-axis, so \( t(x) = 0 \).
To find the x-intercept, solve the equation \( x^2 - x - 2 = 0 \). By factoring, we get \( (x - 2)(x + 1) = 0 \). Solving these factors gives \( x = 2 \) and \( x = -1 \), showing that the x-intercepts are at points (2, 0) and (-1, 0). These points are where the graph touches or passes through the x-axis.
Investigating Y-Intercept
The y-intercept is a fundamental concept in graphing functions. It is the point where the function's graph crosses the y-axis. To find the y-intercept of a rational function, set \( x = 0 \) in the function. This will simplify to the point where the function's output is the y-coordinate:
  • The y-intercept occurs when \( x = 0 \).
  • For our function, \( t(x)=\frac{x^{2}-x-2}{x-6} \), substitute \( x = 0 \).
Plug in zero for \( x \) to get: \( t(0) = \frac{0^2 - 0 - 2}{0 - 6} = \frac{-2}{-6} \). Simplifying, we find the y-intercept at (0, 1/3). This is where the graph intersects the y-axis, providing a point of reference for how the function behaves as it crosses through.
Factoring Quadratics
Factoring quadratic equations is crucial for solving them, especially when working with rational functions. The goal is to rewrite the quadratic in a product form that reveals the solutions. Given the quadratic \( x^2 - x - 2 \), finding its factors involves these steps:
  • If possible, find two numbers that multiply to the constant term (-2) but add up to the middle coefficient (-1).
  • For this equation, the numbers are -2 and 1.
  • Rewrite the quadratic as the product: \( (x - 2)(x + 1) \).
By expressing the quadratic as \( (x - 2)(x + 1) \), we directly see the solutions are \( x = 2 \) and \( x = -1 \). Understanding this process helps in solving for the roots of the equation, which are used to determine x-intercepts efficiently.
Finding Intercepts in Rational Functions
Both x- and y-intercepts provide essential information about a function's graph. For rational functions like \( t(x)=\frac{x^{2}-x-2}{x-6} \), intercepts indicate where the function crosses the axes, giving crucial insight into its behavior:
  • X-intercepts: Occur where the function equals zero, found by setting the numerator zero. Solve \( x^2 - x - 2 = 0 \) to find \( x = 2 \) and \( x = -1 \).
  • Y-intercept: Occurs where \( x = 0 \). Substitute \( x = 0 \) into the function to find it at (0, 1/3).
These intercepts help in sketching the graph and understanding where it may rise or fall relative to the axes. Knowing how to calculate these points makes analyzing and plotting rational functions more straightforward.

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Most popular questions from this chapter

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}}{x^{2}-2}$$

Find all horizontal and vertical asymptotes (if any). $$r(x)=\frac{x^{3}+3 x^{2}}{x^{2}-4}$$

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

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{3 x-x^{2}}{2 x-2}$$

Use a graphing device to find all real solutions of the equation, correct to two decimal places. $$4.00 x^{4}+4.00 x^{3}-10.96 x^{2}-5.88 x+9.09=0$$
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