Question

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, \(y=x^{2}\) ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. $$ f(x)=x^{2}-1 $$

Short answer

The function \(f(x) = x^2 - 1\) is a parabola shifted down by 1 unit. Domain: \((-∞, ∞)\), Range: \([-1, ∞) \).

Step-by-step solution

Identify the Basic Function

The basic function is given by \(y = x^2\). This is a standard parabola that opens upwards with its vertex at the origin (0,0).

Apply the Vertical Shift

The function provided is \(f(x) = x^2 - 1\). This is \(y = x^2\) shifted downward by 1 unit. Each point on the basic graph \(y = x^2\) will move 1 unit down. So, the vertex, which was originally at (0,0), is now at (0,-1).

Plot Key Points

Identify key points on the graph of \(y = x^2\). For example, \((-2, 4)\), \((-1, 1)\), \((0, 0)\), \( (1, 1)\), and \( (2, 4)\). After shifting down by 1 unit, these points become \((-2, 3)\), \((-1, 0)\), \((0, -1)\), \( (1, 0)\), and \( (2, 3)\). Plot these points on the coordinate plane.

Draw the Transformed Graph

Connect the key points with a smooth curve to complete the graph of \(f(x) = x^2 - 1\). The new graph is similar to the original parabola \(y = x^2\) but is shifted downward by 1 unit.

Determine the Domain

The domain of \(f(x) = x^2 - 1\) is all real numbers, since any real x-value can be squared and then have 1 subtracted. In interval notation, the domain is \((-∞, ∞) \).

Determine the Range

The range of \(f(x) = x^2 - 1\) starts from the vertex and extends upwards. The minimum value of \(f(x) \) is -1 (at the vertex), and it increases without bound. In interval notation, the range is \([-1, ∞) \).

Key concepts

Vertical Shift

When we talk about a vertical shift in the context of quadratic functions, we are referring to how the graph of the function moves up or down along the y-axis. For the function given in the exercise,\( f(x) = x^2 - 1 \), the graph is a shifted version of the basic parabola \( y = x^2 \).
In this case, every point on the graph of \( y = x^2 \) is moved 1 unit down.
Let's break it down:

• The original vertex of \( y = x^2 \) is at (0,0).
• After the vertical shift, the new vertex is at (0,-1).

In essence, all points that were originally on the graph are now 1 unit lower. This transformation is crucial when sketching the new graph.
It maintains the shape of the parabola but just translates it downward.
So, if you understand this key point, you're halfway there to accurately graphing the function!

Domain and Range

The domain and range of a function tell us where the function exists in the x-y coordinate plane.
Let's start with the domain. The domain of a quadratic function like \( f(x) = x^2 - 1 \) refers to all possible x-values.
In this case, since any real number can be squared and then have one subtracted from it, the domain includes all real numbers. So, in interval notation, the domain is \((-∞, ∞)\).
Now, let's switch to the range, which describes all possible y-values that the function can take.

• The lowest point on the graph is the new vertex at (0, -1).
• From there, the function opens upwards without any bound.

This means the smallest value is -1, but it can go to positive infinity. Thus, the range in interval notation is \([-1, ∞)\).
Understanding the domain and range is fundamental for fully grasping the behavior of quadratic functions and where they are plotted on the graph.

Parabola Transformation

Transforming a parabola involves changing its position or shape on the graph.
For the given function, we only have a vertical shift, but it's good to know all types of transformations:

• Vertical and horizontal shifts: Moving the graph up, down, left, or right.
• Stretching or compressing: Changing the width of the parabola.
• Reflecting: Flipping the graph over the x-axis or y-axis.

In our example, \( f(x) = x^2 - 1 \), we are dealing with a vertical shift. To transform the basic function \( y = x^2 \) to \( f(x) = x^2 - 1 \):

• Each point on the basic graph moves one unit down.
• The shape and orientation of the parabola stay the same.

Once you grasp how each point is adjusted, transforming and graphing the function becomes straightforward.
Remember, visualizing these transformations can greatly aid your understanding!

Key Points

Identifying key points is a critical step in graphing functions. These points help to accurately plot the graph.
Let’s identify key points for the basic quadratic function \( y = x^2 \):

• For \( x = -2 \), \( y = 4 \).
• For \( x = -1 \), \( y = 1 \).
• For \( x = 0 \), \( y = 0 \) (the vertex).
• For \( x = 1 \), \( y = 1 \).
• For \( x = 2 \), \( y = 4 \).

When we apply the vertical shift to our function \( f(x) = x^2 - 1 \):

• The point \( (-2, 4) \) becomes \( (-2, 3) \).
• The point \( (-1, 1) \) becomes \( (-1, 0) \).
• The point \( (0, 0) \) becomes \( (0, -1) \).
• The point \( (1, 1) \) becomes \( (1, 0) \).
• The point \( (2, 4) \) becomes \( (2, 3) \).

Plot these new points on the graph, and connect them smoothly to visualize the transformed parabola.
Key points give structure and accuracy to your graph, ensuring it is correctly drawn and reflective of the function's behavior.