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Chapter 3: Problem 66

In Exercises \(37-66,\) graph the indicated functions. Plot the graphs of (a) \(y=x^{2}-x+1\) and \((b) y=\frac{x^{3}+1}{x+1}\)

Short Answer

Expert verified
The graph for both functions is a parabola with vertex at \((\frac{1}{2}, \frac{3}{4})\) and a hole at \(x = -1\) for \(y = \frac{x^3 + 1}{x + 1}\).

Step by step solution

01

Analyzing the Quadratic Function

First, consider the quadratic function given by \(y = x^2 - x + 1\). It is a parabola because it is a second-degree polynomial. The coefficient of \(x^2\) is positive, so the parabola opens upwards. You can find its vertex using the vertex formula \(x_v = -\frac{b}{2a}\), where \(a = 1\) and \(b = -1\). Compute the vertex's x-coordinate: \(x_v = -\frac{-1}{2 \times 1} = \frac{1}{2}\). Compute the y-coordinate by substituting \(x_v\) back into the function: \(y_v = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{3}{4}\). Therefore, the vertex is \(\left(\frac{1}{2}, \frac{3}{4}\right)\).
02

Plot the Vertex and Parabola

To graph the quadratic function, plot the vertex at \(\left(\frac{1}{2}, \frac{3}{4}\right)\). Find additional points by selecting values of \(x\), such as \(x = 0\) and \(x = 1\), and calculate corresponding \(y\)-values: For \(x = 0\), \(y = 1\); for \(x = 1\), \(y = 1\). Plot these points as well. Since it's a symmetric parabola, you can reflect these points across the vertex line, e.g., for \(x = -1\), \(y = 3\). Draw a smooth curve through these points to complete the parabola.
03

Simplify the Rational Function

Now consider the function \(y = \frac{x^3 + 1}{x + 1}\). Start by simplifying the expression. Notice that the numerator \(x^3 + 1\) can be factored using the sum of cubes formula \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 1\), so it becomes \((x + 1)(x^2 - x + 1)\). Therefore, \(y = \frac{(x+1)(x^2 - x + 1)}{x + 1}\). Simplify by canceling \(x + 1\) in the numerator and denominator, resulting in \(y = x^2 - x + 1\), for \(x eq -1\).
04

Plot the Simplified Rational Function

The simplified rational function \(y = x^2 - x + 1\) is the same as the quadratic function from Step 1, except it's defined for all \(x\) except \(x = -1\) due to the original division term. At \(x = -1\), plot a hole because the original function is undefined there. For \(x eq -1\), the graph is identical to the parabola plotted in Step 2. Make sure to indicate the hole at \(x = -1\).

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Key Concepts

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

quadratic function
A **quadratic function** is a type of polynomial function characterized by the equation of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Quadratic functions create a parabolic graph that is either U-shaped or n-shaped, depending on the sign of \(a\).
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), the parabola opens downwards.
Quadratic equations are commonly used in various topics within algebra and calculus, and they often appear in situations involving projectile motion and area calculations.
rational function
A **rational function** is a ratio between two polynomial functions. It typically takes the form \(y = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are both polynomials, and \(Q(x)\) is not zero. Rational functions can exhibit a variety of behaviors that are not seen in simpler polynomial functions, such as asymptotes and holes.
  • **Vertical asymptotes** occur where the denominator is zero, but the numerator is not.
  • **Holes** occur where both the numerator and denominator are zero at the same point.
For our exercise, simplifying the function \(y = \frac{x^3 + 1}{x + 1}\) resulted in understanding it as a simplified quadratic except at the point \(x = -1\), illustrating a hole since it makes the original denominator zero.
parabola
The **parabola** is the graph of a quadratic function. It has a unique, symmetric U or n shape. Understanding its features helps in graphing quadratic functions effectively.
Some key features of a parabola include:
  • **Vertex**: The peak or the lowest point of the parabola, where it changes direction.
  • **Axis of symmetry**: A vertical line that divides the parabola into two mirror-image halves. This line always passes through the vertex.
  • **Focus and directrix**: Elements used in geometric definitions of parabolas but less critical for basic plotting.
Knowing the vertex, and using the axis of symmetry, facilitates accurately sketching the curve of the parabola.
vertex formula
The **vertex formula** offers a quick way to identify the vertex of a quadratic function without graphing. The formula is derived from the standard form of a quadratic equation \(y = ax^2 + bx + c\).
To find the vertex:
  • Calculate the x-coordinate using \(x_v = -\frac{b}{2a}\).
  • Substitute \(x_v\) back into the equation to find the y-coordinate: \(y_v = ax_v^2 + bx_v + c\).
In our example with \(y = x^2 - x + 1\), we found the vertex as \(\left(\frac{1}{2}, \frac{3}{4}\right)\). This formula is crucial for quickly determining the most "strategic" point of the parabola which helps in graphing and analyzing the quadratic function.

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Most popular questions from this chapter

Graph the given functions. $$V=e^{3}$$

Use the following table that gives the rate \(R\) of discharge from a tank of water as a function of the height \(H\) of water in the tank. Plot the graph and find the values from the graph. $$\begin{array}{l|c|c|c|c|c|c|c} \text {Height} \text { (ft) } & 0 & 1.0 & 2.0 & 4.0 & 6.0 & 8.0 & 12 \\ \hline \text {Rate }\left(\mathrm{ft}^{3} / \mathrm{s}\right) & 0 & 10 & 15 & 22 & 27 & 31 & 35 \end{array}$$ (a) For \(R=34 \mathrm{ft}^{3} / \mathrm{s}\), find \(H,\) (b) For \(H=6.4 \mathrm{ft},\) find \(R.\)

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