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Chapter 3: Problem 7

For questions 7 and \(8,\) do not use your GDC. a) Given \(y=2 x^{3}+3 x^{2}-5 x-4,\) determine the \(y\) -value for each value of \(x\) such that \(x \in\\{-3,-2,-1,0,1,2,3\\}\) b) How many times must the graph of \(y=2 x^{3}+3 x^{2}-5 x-4\) cross the \(x\) -axis? c) Sketch the graph of \(y=2 x^{3}+3 x^{2}-5 x-4\)

Short Answer

Expert verified
a) y-values: -16, 2, 2, -4, -4, 14, 62. b) The graph crosses the x-axis once. c) Cubic curve starting steep negative, aligning through plotted y-values.

Step by step solution

01

Evaluate the Function for Given x-values

The function is given by \( y = 2x^3 + 3x^2 - 5x - 4 \). Evaluate \( y \) for each \( x \) in \( \{-3,-2,-1,0,1,2,3\} \).- For \( x = -3 \): \( y = 2(-3)^3 + 3(-3)^2 - 5(-3) - 4 \) \[ y = 2(-27) + 3(9) + 15 - 4 = -54 + 27 + 15 - 4 = -16 \]- For \( x = -2 \): \( y = 2(-2)^3 + 3(-2)^2 - 5(-2) - 4 \) \[ y = 2(-8) + 3(4) + 10 - 4 = -16 + 12 + 10 - 4 = 2 \]- For \( x = -1 \): \( y = 2(-1)^3 + 3(-1)^2 - 5(-1) - 4 \) \[ y = 2(-1) + 3(1) + 5 - 4 = -2 + 3 + 5 - 4 = 2 \]- For \( x = 0 \): \( y = 2(0)^3 + 3(0)^2 - 5(0) - 4 \) \[ y = -4 \]- For \( x = 1 \): \( y = 2(1)^3 + 3(1)^2 - 5(1) - 4 \) \[ y = 2 + 3 - 5 - 4 = -4 \]- For \( x = 2 \): \( y = 2(2)^3 + 3(2)^2 - 5(2) - 4 \) \[ y = 16 + 12 - 10 - 4 = 14 \]- For \( x = 3 \): \( y = 2(3)^3 + 3(3)^2 - 5(3) - 4 \) \[ y = 54 + 27 - 15 - 4 = 62 \].
02

Determine the x-intercepts

To find where the graph crosses the \( x \)-axis, set \( y = 0 \) and solve the equation.Given: \( 2x^3 + 3x^2 - 5x - 4 = 0 \).Finding the roots for a cubic polynomial analytically can be complex, but it generally has at least one real root due to the Intermediate Value Theorem. In this case, factoring or using polynomial division if needed to simplify the solution. Check values around zero: test integers first, find potential roots and use synthetic division or direct substitution to confirm.For balanced approximation (plot-based), turns out:- \( x = 1 \) yields \( y = -4 \) as incorrect guess, hence try others. - Derive approximation solutions through calculations, hence graph relies on numeric approximation.Identify 1 root as \( x = -3 \) presents approximated limit for sign-changing.
03

Sketch the Graph

Using the evaluated y-values from Step 1:1. Plot the points \((-3, -16)\), \((-2, 2)\), \((-1, 2)\), \((0, -4)\), \((1, -4)\), \((2, 14)\), \((3, 62)\) on the graph.2. Note the curve behavior through observation of examined function for continuity and change in function slope.3. Sketch the cubic nature beginning steep (negative) towards moderate upward trend, reflecting crossings or extremes as verified results demonstrate two changes below \( x = 0 \).4. Mark where curve crosses or touches the x-axis based on discovery feature of action and change in y-values.

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Key Concepts

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

Polynomial Evaluation
Evaluating a polynomial function is a simple yet crucial step in understanding its behavior. Polynomial evaluation means calculating the value of a polynomial expression for a specific value of the variable. For example, given the cubic function \(y = 2x^3 + 3x^2 - 5x - 4\), to evaluate it for a specific \(x\)-value like \(-3\), follow these steps:
  • Substitute the \(x\)-value into the polynomial. For \(x = -3\), it becomes: \(y = 2(-3)^3 + 3(-3)^2 - 5(-3) - 4\).
  • Calculate each term: \(2(-27) + 3(9) + 15 - 4\).
  • Add/subtract the results: \(-54 + 27 + 15 - 4 = -16\).
This process reveals the \(y\)-value for the given \(x\), helping you gain insights into the function's shape and overall behavior at various points.
Graph Sketching
Graph sketching of polynomials involves plotting several points and understanding the general shape of the graph. With cubic functions like \(y = 2x^3 + 3x^2 - 5x - 4\), the graph exhibits typical cubic behavior:
  • Identify key points by evaluating the polynomial for a range of \(x\)-values. For example, points \((-3, -16)\), \((0, -4)\), and \((3, 62)\) provide critical information.
  • Note the end behavior. Cubic functions have distinct end behavior: one end goes to infinity and the other to negative infinity.
  • Observe increases and decreases in \(y\)-values to see where the function slopes up or down.
The graph of our example starts with a steep negative slope then rises, showing its roots and turning points, shaping a typical cubic curve. This can help visualize the polynomial鈥檚 interaction with the \(x\)-axis and the curvature away from it.
Intermediate Value Theorem
The Intermediate Value Theorem is a powerful tool for analyzing continuous functions like polynomials. Simply put, it states that for any value \(k\) between \(f(a)\) and \(f(b)\), there exists a \(c\) in the interval \([a, b]\) such that \(f(c) = k\). This is crucial when locating roots of polynomials:
  • If \(f(x)\) changes sign between two values, then there is at least one root in that interval.
  • In the cubic function \(y = 2x^3 + 3x^2 - 5x - 4\), if there's a sign change between evaluated points, a root exists.
  • Explore the intervals where \(y\) values switch from positive to negative or vice versa.
By applying this theorem, it becomes easier to approximate the location of roots without exact calculations, especially beneficial for more complex polynomials.
Finding Roots of Polynomials
Determining roots of a polynomial involves finding the \(x\)-values where the polynomial equals zero. For cubic equations, this can be quite challenging but is essential in sketching accurate graphs and modeling behavior:
  • Set the polynomial equal to zero, like \(2x^3 + 3x^2 - 5x - 4 = 0\).
  • Using tools such as factoring, polynomial division, or synthetic division simplifies the polynomial to reveal roots.
  • For the function stated, trials of integer values like \(x = -3\) showed potential roots based on evaluation.
Identifying these values is integral in structuring the graph as they represent where the curve intersects the \(x\)-axis. Successfully finding roots aids in better comprehension of all polynomial functions.

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Most popular questions from this chapter

Decompose the following rational expressions into partial fractions. $$\frac{12}{x^{4}-x^{3}-2 x^{2}}$$

Decompose the following rational expressions into partial fractions. $$\frac{3 x+2}{x^{3}+6 x}$$

Find a polynomial of lowest degree with real coefficients and the given zeros. $$x=2, x=-4 \text { and } x=-3 i$$

Show that both of the following inequalities are true for all real numbers \(m\) and \(n\) such that \(m>n>0\) a) \(m+\frac{1}{n}>2\) b) \((m+n)\left(\frac{1}{m}+\frac{1}{n}\right)>4\)

Sketch the graph of the rational function without the aid of your GDC. On your sketch clearly indicate any \(x\) - or \(y\) -intercepts and any asymptotes (vertical, horizontal or oblique). Use your GDC to verify your sketch. $$g(x)=\frac{3}{x-2}$$
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