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Chapter 3: Problem 9

Graph the following on the same coordinate system. (a) \(y=x^{2}\) (b) \(y=3 x^{2}\) (c) \(y=\frac{1}{3} x^{2}\) (d) How does the coefficient of \(x^{2}\) affect the shape of the graph?

Short Answer

Expert verified
The coefficient of \( x^2 \) affects the width of the parabola: larger makes it narrower, smaller makes it wider.

Step by step solution

01

Graph the first equation

Graph the equation \( y = x^2 \) on the coordinate system. This is a standard parabola with the vertex at the origin (0,0) and opens upwards.
02

Graph the second equation

Graph the equation \( y = 3x^2 \). Notice that this parabola will also have its vertex at the origin but will be narrower than \( y = x^2 \) due to the greater coefficient of \( x^2 \).
03

Graph the third equation

Graph the equation \( y = \frac{1}{3}x^2 \). This parabola will also have its vertex at the origin, but it will be wider than \( y = x^2 \) due to the smaller coefficient of \( x^2 \).
04

Compare the graphs

Observe and compare the shapes of all three parabolas. \( y = x^2 \) is the standard parabola, \( y = 3x^2 \) is narrower, and \( y = \frac{1}{3}x^2 \) is wider.
05

Analyze the effect of the coefficient

The coefficient of \( x^2 \) determines the width of the parabola. A larger coefficient makes the parabola narrower, while a smaller coefficient makes it wider.

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Key Concepts

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

Parabolas
When you graph the function of a quadratic equation, such as those given in the exercise, the resulting shape is called a parabola. Parabolas have a unique, symmetric 'U' shape. Here are a few important features:
  • They are always symmetrical around a vertical line called the axis of symmetry. For standard parabolas that open upwards or downwards, this line goes through the vertex.
  • They have a highest or lowest point known as the vertex.
  • The direction in which the parabola opens (upwards or downwards) depends on the coefficient of the quadratic term.

Understanding these aspects will help you when trying to graph and compare different quadratic functions. Each parabola can look different depending on its equation, making it important to carefully examine the coefficients.
Vertex
The vertex is a key feature of a parabola. It鈥檚 either the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards, respectively. For a standard quadratic equation in the form of:
\ y = ax^2 + bx + c \
The vertex can be found using the formula for its coordinates:
\ x = -b / (2a) \
However, in the given exercise with simplified equations like:
  • y = x^2
  • y = 3x^2
  • y = \( \frac{1}{3}x^2 \)

The vertex is at the origin (0,0). That means all the given parabolas share the same vertex. Noticing the position of the vertex when graphing will always be helpful, as it allows you to quickly identify the shape and orientation of the parabola.
Coefficient Effect
The coefficients in a quadratic equation greatly influence the shape of the corresponding parabola. Specifically, the coefficient of \( x^2 \), often denoted as 'a' in the general form \( y = ax^2 \), determines how wide or narrow the parabola appears.
  • A larger coefficient (greater |a| value) makes the parabola narrower. This is seen in the equation \( y = 3x^2 \).
  • A smaller coefficient (lesser |a| value) makes the parabola wider. This is illustrated by \( y = \frac{1}{3}x^2 \).
  • If the coefficient is positive, the parabola opens upwards. If it is negative, the parabola opens downwards (not covered in this exercise but important to note).

This effect can be visualized by comparing the graphs of the given equations.

Example:
  • For \( y = x^2 \), a = 1, giving a standard 'U' shape.
  • For \( y = 3x^2 \), a = 3, resulting in a narrower, steeper parabola.
  • For \( y = \frac{1}{3}x^2 \), a = 1/3, leading to a wider, more gently curving parabola.

By analyzing these coefficient effects, you can better predict and understand how changes in the equation affect the graph.

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Most popular questions from this chapter

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{4}+2 x^{3}-3 x^{2}+24 x-180$$

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. Given that \(-4\) and \(-1\) are zeros of the numerator, factor the numerator completely.

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. What is the \(y\) -intercept of the graph of \(f ?\)

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. (a) Given that 1 and 2 are zeros of the denominator, factor the denominator completely. (b) Write the entire quotient for \(f\) so that the numerator and the denominator are in factored form.

See Exercise \(117 .\) Suppose an economist determines that $$R(x)=\frac{60 x-6000}{x-120}$$ where \(y=R(x)\) is government revenue in tens of millions of dollars for a tax rate of \(x\) percent, with \(y=R(x)\) valid for \(50 \leq x \leq 100 .\) Find the revenue for each tax rate. (a) \(50 \%\) (b) \(60 \%\) (c) \(80 \%\) (d) \(100 \%\) (e) Graph \(R\) in the window \([0,100]\) by \([0,50]\)
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