Question

determine whether each statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement.

the graph of (f(x) = -4(x + 3)^2 - 2) has one y-intercept and two x-intercepts.

choose the correct answer below.

a. the statement is true.
b. the statement is false. a true statement is \the graph of (f(x) = -4(x + 3)^2 - 2) has no y-intercept and no x-intercepts.\
c. the statement is false. a true statement is \the graph of (f(x) = -4(x + 3)^2 - 2) has one y-intercept and one x-intercept.\
d. the statement is false. a true statement is \the graph of (f(x) = -4(x + 3)^2 - 2) has one y-intercept and no x-intercepts.\

Explanation:

Identify the vertex and opening direction

Using the Vertex Form of a Quadratic knowledge point

$$
LATEXBLOCK0
$$

Determine the number of x-intercepts

Using the Parabola Graphing knowledge point

$$
LATEXBLOCK1
$$

Determine the number of y-intercepts

Every vertical parabola of the form \(f(x) = ax^2 + bx + c\) has exactly one \(y\)-intercept, which is found by evaluating \(f(0)\):

$$ f(0) = -4(0+3)^2 - 2 = -4(9) - 2 = -38 $$

Thus, there is exactly one \(y\)-intercept.

Evaluate the statements

The original statement claims the graph has "one \(y\)-intercept and two \(x\)-intercepts," which is false.
The correct statement is: "The graph of \(f(x) = -4(x+3)^2 - 2\) has one \(y\)-intercept and no \(x\)-intercepts."
This matches option D.

Answer:

• A. The statement is true.
• B. The statement is false. A true statement is "The graph of \(f(x) = -4(x+3)^2 - 2\) has no y-intercept and no x-intercepts."
• C. The statement is false. A true statement is "The graph of \(f(x) = -4(x+3)^2 - 2\) has one y-intercept and one x-intercept."
• D. The statement is false. A true statement is "The graph of \(f(x) = -4(x+3)^2 - 2\) has one y-intercept and no x-intercepts." (Correct answer)