Problem 10

Question

Sketch the graph of the function and compare it with the graph of $y=x^{2}$ $y=x^{2}-1$

Step-by-Step Solution

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Answer

The graph of the function $y=x^{2}-1$ is a parabola, similar to the graph of $y=x^{2}$, but translated down by 1 unit.

1Step 1: understand the transformation

Notice that the function $y=x^{2}-1$ is the function $y=x^{2}$ shifted down by 1 unit. This is called a vertical translation.

2Step 2: sketch the basic function

Draw the graph of $y=x^{2}$, which is a parabola. The parabola is symmetric with respect to the y-axis (called the axis of symmetry), opens upward, and its vertex is at the origin (0,0).

3Step 3: sketch the transformed function

Using the graph from step 2 as a reference, shift the entire graph down by 1 unit to get the graph of $y=x^{2}-1$. The new graph is still a parabola, still symmetric with respect to the y-axis but its vertex is now at (0, -1).

Key Concepts

Graph TransformationsParabolasVertical Translation

Graph Transformations

Understanding graph transformations is key when working with functions like quadratics. By transforming a graph, you are essentially modifying its position, shape, or size on the coordinate plane.

Transformations affect the entire graph of a function.
Types include translations (shifts), reflections, dilations, and rotations.
A transformation can move a graph around or stretch/compress it.

For quadratic functions like $y = x^2$, you often encounter translations, as in the function $y = x^2 - 1$. Here, the graph is shifted down by 1 unit due to the "-1" term, a vertical translation. Recognizing these transformations helps in quickly sketching and understanding complex functions.

Parabolas

Quadratic functions create a specific curve known as a parabola. A parabola is distinctive and has several easily identifiable properties.

It is a symmetrical curve.
Has a U-shape that can open upward or downward.
The graph of $y = x^2$ is a standard upward-opening parabola.

The highest or lowest point on a parabola is called the vertex. For $y = x^2$, the vertex is at the origin, (0,0). Its axis of symmetry is the y-axis, meaning the parabola is mirrored perfectly on both sides of this line. Understanding these basic attributes will help you analyze more complex quadratics by breaking them into simpler, understandable parts.

Vertical Translation

Vertical translation involves shifting a graph up or down on the coordinate plane. This type of transformation is quite common with functions of all types.

Translation doesn't alter the shape, only the position.
Addition or subtraction outside the function's formula shifts its graph vertically.

For example, the quadratic function $y = x^2 - 1$ undergoes a vertical translation. Starting with $y = x^2$, subtracting 1 shifts the entire graph down by one unit. Notice this does not affect the parabola's symmetry or opening direction, only its vertical position. This concept is crucial, as it helps you visualize how shifts impact where the graph sits in your coordinate system.

Problem 10

Problem 11

Other exercises in this chapter

Problem 10

Confirm that the function has the indicated zeros. $$f(x)=x^{3}+9 x ; 0,-3 i, 3 i$$

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Problem 10

Use long division to divide and use the result to factor the dividend completely. $$\left(5 x^{2}-17 x-12\right) \div(x-4)$$

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Problem 11

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model,

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Problem 11

Use a graphing utility to graph $f(x)=2 / x$ and the function $g$ in the same viewing window. Describe the relationship between the two graphs. $$g(x)=-f(x)

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