Math / Algebra

QuestionConsider the function $f(x)=3 x^{2}-24 x-8$ . a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

Studdy Solution

STEP 1

1. The function $f(x) = 3x^2 - 24x - 8$ is a quadratic function.
2. Quadratic functions are of the form $ax^2 + bx + c$ and have a parabolic shape.
3. The sign of the coefficient $a$ determines if the parabola opens upwards or downwards.
4. The vertex of the parabola gives the minimum or maximum value of the function.
5. The domain of a quadratic function is all real numbers.
6. The range depends on whether the function has a minimum or maximum value.

STEP 2

1. Determine if the function has a minimum or maximum value.
2. Find the vertex of the parabola to determine the minimum or maximum value and where it occurs.
3. Identify the domain and range of the function.

STEP 3

To determine if the function has a minimum or maximum value, examine the coefficient of $x^2$ in the quadratic function $f(x) = 3x^2 - 24x - 8$ .
The coefficient $a = 3$ is positive, which means the parabola opens upwards. Therefore, the function has a minimum value.

STEP 4

To find the minimum value and where it occurs, calculate the vertex of the parabola. The x-coordinate of the vertex for a quadratic function $ax^2 + bx + c$ is given by:
$x = -\frac{b}{2a}$
For $f(x) = 3x^2 - 24x - 8$ , $a = 3$ and $b = -24$ .
$x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4$

STEP 5

Substitute $x = 4$ back into the function to find the minimum value:
$f(4) = 3(4)^2 - 24(4) - 8$ $f(4) = 3(16) - 96 - 8$ $f(4) = 48 - 96 - 8$ $f(4) = -56$
The minimum value is $-56$ and it occurs at $x = 4$ .

STEP 6

Identify the domain of the function. Since $f(x) = 3x^2 - 24x - 8$ is a quadratic function, its domain is all real numbers:
$\text{Domain: } (-\infty, \infty)$

STEP 7

Identify the range of the function. Since the function has a minimum value of $-56$ at $x = 4$ and the parabola opens upwards, the range is:
$\text{Range: } [-56, \infty)$
The function has a minimum value of $-56$ at $x = 4$ , the domain is $(-\infty, \infty)$ , and the range is $[-56, \infty)$ .

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QuestionConsider the function $f(x)=3 x^{2}-24 x-8$ . a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

Studdy Solution

STEP 1

STEP 2

1. Determine if the function has a minimum or maximum value.
2. Find the vertex of the parabola to determine the minimum or maximum value and where it occurs.
3. Identify the domain and range of the function.

STEP 3

To determine if the function has a minimum or maximum value, examine the coefficient of $x^2$ in the quadratic function $f(x) = 3x^2 - 24x - 8$ .
The coefficient $a = 3$ is positive, which means the parabola opens upwards. Therefore, the function has a minimum value.

STEP 4

To find the minimum value and where it occurs, calculate the vertex of the parabola. The x-coordinate of the vertex for a quadratic function $ax^2 + bx + c$ is given by:
$x = -\frac{b}{2a}$
For $f(x) = 3x^2 - 24x - 8$ , $a = 3$ and $b = -24$ .
$x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4$

STEP 5

Substitute $x = 4$ back into the function to find the minimum value:
$f(4) = 3(4)^2 - 24(4) - 8$ $f(4) = 3(16) - 96 - 8$ $f(4) = 48 - 96 - 8$ $f(4) = -56$
The minimum value is $-56$ and it occurs at $x = 4$ .

STEP 6

Identify the domain of the function. Since $f(x) = 3x^2 - 24x - 8$ is a quadratic function, its domain is all real numbers:
$\text{Domain: } (-\infty, \infty)$

STEP 7

Identify the range of the function. Since the function has a minimum value of $-56$ at $x = 4$ and the parabola opens upwards, the range is:
$\text{Range: } [-56, \infty)$
The function has a minimum value of $-56$ at $x = 4$ , the domain is $(-\infty, \infty)$ , and the range is $[-56, \infty)$ .

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QuestionConsider the function f(x)=3x2−24x−8f(x)=3 x^{2}-24 x-8f(x)=3x2−24x−8. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

Studdy Solution

STEP 1

STEP 2

1. Determine if the function has a minimum or maximum value.2. Find the vertex of the parabola to determine the minimum or maximum value and where it occurs.3. Identify the domain and range of the function.

STEP 3

STEP 4

STEP 5

STEP 6

Identify the domain of the function. Since f(x)=3x2−24x−8 f(x) = 3x^2 - 24x - 8 f(x)=3x2−24x−8 is a quadratic function, its domain is all real numbers:Domain: (−∞,∞) \text{Domain: } (-\infty, \infty) Domain: (−∞,∞)

STEP 7

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionConsider the function $f(x)=3 x^{2}-24 x-8$ . a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

1. Determine if the function has a minimum or maximum value.
2. Find the vertex of the parabola to determine the minimum or maximum value and where it occurs.
3. Identify the domain and range of the function.

Identify the domain of the function. Since $f(x) = 3x^2 - 24x - 8$ is a quadratic function, its domain is all real numbers:
$\text{Domain: } (-\infty, \infty)$