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Chapter 3: Problem 72

Use a graphing utility. Graph \(y=x^{2}, y=x^{4},\) and \(y=x^{6}\) on the same screen. What do you notice is the same about each graph? What do you notice is different?

Short Answer

Expert verified
All graphs are parabolas opening upwards with vertices at (0,0). Higher exponents make the graphs narrower and steeper.

Step by step solution

01

- Graph the Function y=x^2

Using a graphing utility, plot the function \( y = x^2 \). This is a parabola opening upwards with its vertex at the origin (0,0). Note the shape and the width of the curve.
02

- Graph the Function y=x^4

Next, plot the function \( y = x^4 \) on the same graph. This is also a parabola opening upwards, but it is narrower than \( y = x^2 \) and has a sharper turn at the origin.
03

- Graph the Function y=x^6

Finally, plot the function \( y = x^6 \). This graph is similar in shape to the previous two, but even narrower and with an even sharper turn at the origin.
04

- Compare the Graphs

Observe the similarities and differences between the graphs of the three functions. All three graphs are parabolas opening upwards and symmetric about the y-axis, with vertices at (0,0). The main difference is that as the exponent increases (from 2 to 6), the graph becomes narrower and steeper.

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Key Concepts

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

Plotting Functions
Plotting functions is a vital skill in understanding how different functions behave. When you plot the graph of a function, you are converting an algebraic expression into a visual form.
To visualize a function, you can use graphing utilities or software like Desmos or a graphing calculator. Start by identifying key points and aspects, such as intercepts, symmetry, and general shape.
For example, when plotting the function \( y = x^2 \), input the function into your graphing tool and observe the output. You'll see a parabola opening upwards, centered at the origin. Repeat this process for other functions to compare their shapes easily.
Graph Transformations
Graph transformations involve altering the position or shape of a graph. These transformations can be shifts, reflections, stretches, or compressions.
When shifting a graph, you're moving it up, down, left, or right. For instance, adding a positive constant to \( y = x^2 \) shifts the graph upwards, while subtracting shifts it downwards.
  • Reflection: Flipping the graph over an axis.
  • Stretching/Compressing: Making the graph wider or narrower.
  • Rotation: Rotating the graph around a point.
For polynomial functions, increasing the exponent results in the graph becoming narrower and steeper. This is a visual example of compression.
Polynomial Graphs
Polynomial graphs show the relationship expressed by polynomial equations. A polynomial function is usually written in the form \( y = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \).
These graphs can exhibit various shapes, but higher-degree polynomials often form U-shaped curves called parabolas.
For instance, when graphing \( y = x^2 \), the resulting parabola opens upwards. For \( y = x^4 \) and \( y = x^6 \), you'll notice similar U-shaped graphs but narrower as the exponent increases. This means the higher the degree of the polynomial, the steeper the graph near the origin.
Understanding these patterns help in predicting the behavior of more complex polynomial functions.

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