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Chapter 3: Problem 3

Find the intercepts of the equation \(y=x^{3}-8\)

Short Answer

Expert verified
The y-intercept is (0, -8) and the x-intercept is (2, 0).

Step by step solution

01

Find the y-intercept (when x=0)

To find the y-intercept, set \( x = 0 \) in the equation. \[ y = 0^3 - 8 \] Simplify the equation: \[ y = -8 \] So, the y-intercept is (0, -8).
02

Find the x-intercepts (when y=0)

To find the x-intercepts, set \( y = 0 \) in the equation: \[ 0 = x^3 - 8 \] Solve for \( x \): \[ x^3 = 8 \] Take the cube root of both sides: \[ x = \root 3 otemath{(8)} \] So, the x-intercept is (2, 0).

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Key Concepts

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

Understanding the Y-Intercept
The y-intercept of a function is where the graph intersects the y-axis. This happens when the value of x is 0. To find the y-intercept for the equation given, set x to 0 and solve for y. In this example, inserting x = 0 into the equation, we get:\[ y = 0^3 - 8 \ y = -8 \]So, the y-intercept is (0, -8). This means the graph crosses the y-axis at -8.
Finding the X-Intercepts
The x-intercepts occur where the graph intersects the x-axis. This is when y equals 0. To find the x-intercepts for the equation, set y to 0 and solve for x. For the equation \( y = x^3 - 8 \):\[ 0 = x^3 - 8 \]Add 8 to both sides to isolate the cubic term:\[ x^3 = 8 \]Now, take the cube root of both sides:\[ x = \root 3 \relax{(8)} = 2 \]So, the x-intercept is (2, 0). The graph of this function crosses the x-axis at x = 2.
Solving Equations for Intercepts
Solving equations involves finding the values of variables that satisfy the equation. For intercepts, this usually means setting the opposite variable to 0 and solving.
  • For the y-intercept, set x = 0 and solve for y.
  • For the x-intercepts, set y = 0 and solve for x.
In the given problem, we used these methods to find:
  • Y-intercept: (0, -8)
  • X-intercept: (2, 0)
This consistent approach makes finding intercepts straightforward.
Polynomial Functions and Their Intercepts
Polynomial functions are expressions that involve sums of powers of variables, like \(x^3 - 8\). These functions can have multiple intercepts depending on their degree (the highest power of the variable). For the cubic function given (which is a polynomial of degree 3), we found one x-intercept and one y-intercept. The nature of polynomial functions is that they can curve and twist, so you could have more complex behaviors. However, the intercepts are points that tell us where these curves cross the axes in a coordinate plane, giving crucial insights about the general shape and position of the function's graph.

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Most popular questions from this chapter

How many \(x\) -intercepts can a function defined on an interval have if it is increasing on that interval? Explain.

The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. Problems \(85-92\) require the following discussion of a secant line. The slope of the secant line containing the two points \((x, f(x))\) and \((x+h, f(x+h))\) on the graph of a function \(y=f(x)\) may be given as \(m_{\mathrm{sec}}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h} \quad h \neq 0\) (a) Express the slope of the secant line of each function in terms of \(x\) and \(h\). Be sure to simplify your answer. (b) Find \(m_{\text {sec }}\) for \(h=0.5,0.1\), and 0.01 at \(x=1 .\) What value does \(m_{\text {sec }}\) approach as h approaches \(0 ?\) (c) Find an equation for the secant line at \(x=1\) with \(h=0.01\). (d) Use a graphing utility to graph fand the secant line found in part ( \(c\) ) in the same viewing window. \(f(x)=x^{2}+2 x\)

Determine algebraically whether each function is even, odd, or neither. \(F(x)=\frac{2 x}{|x|}\)

\(g(x)=x^{2}-2\) (a) Find the average rate of change from -2 to 1 . (b) Find an equation of the secant line containing \((-2, g(-2))\) and \((1, g(1))\).

Find the domain of \(h(x)=\sqrt[4]{x+7}+7 x\).
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