Question

Sketch the graph using transformations. Identify the intercepts, odd/even/neither, decreasing/increasing intervals, end behavior, and domain/range of each.

f(x) = (x+4)²-1

Answer 1

The graph of the function f(x) = (x + 4)² - 1 is added as an attachment and the properties are

• x-intercepts = -5 and -3; y-intercept = 15;
• It is neither odd nor even
• The domain is (-∝, ∝) and the range is f(x) ≥ -1
• The decreasing interval is (-∝, -4) and the increasing interval is (-4, ∝)

Sketching the graph of the function

From the question, we have the following parameters that can be used in our computation:

f(x) = (x + 4)² - 1

The above function is a quadratic function that has been transformed as follows

This means that the domain is the set of all real values and it is neither odd nor even

For the range, we set the squared expression to 0

So, we have

f(x) ≥ 0 - 1

f(x) ≥ -1

The vertex of the function is (-4, -1)

Because the leading coefficient is positive, the decreasing interval is (-∝, -4) and the increasing interval is (-4, ∝)

For the x-intercept, we set the function to 0 and solve for x

So, we have

(x + 4)² - 1 = 0

(x + 4)² = 1

x + 4 = ±1

Evaluate

x-intercepts = -5 and -3

For the y-intercept, we set the x to 0 and solve for y

So, we have

f(0) = (0 + 4)² - 1

f(0) = 15

Next, we plot the graph using a graphing tool by taking not of the above features

The graph of the function is added as an attachment