Math / Algebra

QuestionJetermine 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 $f(x)=-3(x+3)^{2}-6$ has one $y$ -intercept and two $x$ -intercepts.

Studdy Solution

STEP 1

1. The function given is $f(x) = -3(x+3)^2 - 6$ .
2. We need to determine the number of $y$ -intercepts and $x$ -intercepts.
3. A $y$ -intercept occurs where the graph crosses the $y$ -axis, i.e., at $x = 0$ .
4. An $x$ -intercept occurs where the graph crosses the $x$ -axis, i.e., where $f(x) = 0$ .

STEP 2

1. Determine the $y$ -intercept of the function.
2. Determine the $x$ -intercepts of the function.
3. Evaluate the truth of the statement.
4. Correct the statement if necessary.

STEP 3

To find the $y$ -intercept, substitute $x = 0$ into the function:
$f(0) = -3(0+3)^2 - 6$ $f(0) = -3(3)^2 - 6$ $f(0) = -3 \times 9 - 6$ $f(0) = -27 - 6$ $f(0) = -33$
The $y$ -intercept is at $(0, -33)$ .

STEP 4

To find the $x$ -intercepts, set $f(x) = 0$ and solve for $x$ :
$0 = -3(x+3)^2 - 6$ $3(x+3)^2 = -6$ $(x+3)^2 = -2$
Since the square of a real number cannot be negative, there are no real solutions for $x$ . Therefore, there are no $x$ -intercepts.

STEP 5

Evaluate the truth of the statement: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and two $x$ -intercepts."
The statement is false. The graph has one $y$ -intercept and no $x$ -intercepts.

STEP 6

Correct the statement to: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and no $x$ -intercepts."
The corrected statement is: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and no $x$ -intercepts."

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QuestionJetermine 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 $f(x)=-3(x+3)^{2}-6$ has one $y$ -intercept and two $x$ -intercepts.

Studdy Solution

STEP 1

STEP 2

1. Determine the $y$ -intercept of the function.
2. Determine the $x$ -intercepts of the function.
3. Evaluate the truth of the statement.
4. Correct the statement if necessary.

STEP 3

To find the $y$ -intercept, substitute $x = 0$ into the function:
$f(0) = -3(0+3)^2 - 6$ $f(0) = -3(3)^2 - 6$ $f(0) = -3 \times 9 - 6$ $f(0) = -27 - 6$ $f(0) = -33$
The $y$ -intercept is at $(0, -33)$ .

STEP 4

To find the $x$ -intercepts, set $f(x) = 0$ and solve for $x$ :
$0 = -3(x+3)^2 - 6$ $3(x+3)^2 = -6$ $(x+3)^2 = -2$
Since the square of a real number cannot be negative, there are no real solutions for $x$ . Therefore, there are no $x$ -intercepts.

STEP 5

Evaluate the truth of the statement: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and two $x$ -intercepts."
The statement is false. The graph has one $y$ -intercept and no $x$ -intercepts.

STEP 6

Correct the statement to: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and no $x$ -intercepts."
The corrected statement is: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and no $x$ -intercepts."

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QuestionJetermine 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)=−3(x+3)2−6f(x)=-3(x+3)^{2}-6f(x)=−3(x+3)2−6 has one yyy-intercept and two xxx-intercepts.

Studdy Solution

STEP 1

STEP 2

1. Determine the y y y-intercept of the function.2. Determine the x x x-intercepts of the function.3. Evaluate the truth of the statement.4. Correct the statement if necessary.

STEP 3

STEP 4

STEP 5

Evaluate the truth of the statement: "The graph of f(x)=−3(x+3)2−6 f(x) = -3(x+3)^2 - 6 f(x)=−3(x+3)2−6 has one y y y-intercept and two x x x-intercepts."The statement is false. The graph has one y y y-intercept and no x x x-intercepts.

STEP 6

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QuestionJetermine 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 $f(x)=-3(x+3)^{2}-6$ has one $y$ -intercept and two $x$ -intercepts.

1. Determine the $y$ -intercept of the function.
2. Determine the $x$ -intercepts of the function.
3. Evaluate the truth of the statement.
4. Correct the statement if necessary.

Evaluate the truth of the statement: "The graph of $f(x) = -3(x+3)^2 - 6$ has one $y$ -intercept and two $x$ -intercepts."
The statement is false. The graph has one $y$ -intercept and no $x$ -intercepts.