QuestionDetermine if the quadratic function $f(x)=3x^{2}+18x-9$ has a maximum or minimum value and find that value.

Studdy Solution

STEP 1

Assumptions1. The given function is a quadratic function, which is a polynomial of degree. . The function is in the form $f(x) = ax^ + bx + c$ , where $a$ , $b$ , and $c$ are constants.
3. The function has a maximum or minimum value at its vertex.

STEP 2

The coefficient of $x^2$ in the quadratic function determines whether the function has a maximum or minimum value. If the coefficient of $x^2$ is positive, the function has a minimum value. If the coefficient of $x^2$ is negative, the function has a maximum value.

STEP 3

Identify the coefficient of $x^2$ in the given function.
$f(x) =3x^2 +18x -9$ The coefficient of $x^2$ is3.

STEP 4

Since the coefficient of $x^2$ is positive, the function has a minimum value.

STEP 5

To find the minimum value, we first need to find the x-coordinate of the vertex. The x-coordinate of the vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

STEP 6

Substitute the values of $a$ and $b$ into the formula to find the x-coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{18}{2 \cdot3}$

STEP 7

Calculate the x-coordinate of the vertex.
$x = -\frac{18}{6} = -3$

STEP 8

Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting this x-coordinate back into the function.
$f(x) =3(-3)^2 +18(-3) -$

STEP 9

Calculate the minimum value of the function.
$f(-3) =3 \cdot9 -54 -9 =27 -54 -9 = -36$ The function $f$ has a minimum value of -36.

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QuestionDetermine if the quadratic function $f(x)=3x^{2}+18x-9$ has a maximum or minimum value and find that value.

Studdy Solution

STEP 1

Assumptions1. The given function is a quadratic function, which is a polynomial of degree. . The function is in the form $f(x) = ax^ + bx + c$ , where $a$ , $b$ , and $c$ are constants.
3. The function has a maximum or minimum value at its vertex.

STEP 2

The coefficient of $x^2$ in the quadratic function determines whether the function has a maximum or minimum value. If the coefficient of $x^2$ is positive, the function has a minimum value. If the coefficient of $x^2$ is negative, the function has a maximum value.

STEP 3

Identify the coefficient of $x^2$ in the given function.
$f(x) =3x^2 +18x -9$ The coefficient of $x^2$ is3.

STEP 4

Since the coefficient of $x^2$ is positive, the function has a minimum value.

STEP 5

To find the minimum value, we first need to find the x-coordinate of the vertex. The x-coordinate of the vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

STEP 6

Substitute the values of $a$ and $b$ into the formula to find the x-coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{18}{2 \cdot3}$

STEP 7

Calculate the x-coordinate of the vertex.
$x = -\frac{18}{6} = -3$

STEP 8

Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting this x-coordinate back into the function.
$f(x) =3(-3)^2 +18(-3) -$

STEP 9

Calculate the minimum value of the function.
$f(-3) =3 \cdot9 -54 -9 =27 -54 -9 = -36$ The function $f$ has a minimum value of -36.

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QuestionDetermine if the quadratic function f(x)=3x2+18x−9f(x)=3x^{2}+18x-9f(x)=3x2+18x−9 has a maximum or minimum value and find that value.

Studdy Solution

STEP 1

Assumptions1. The given function is a quadratic function, which is a polynomial of degree. . The function is in the form f(x)=ax+bx+cf(x) = ax^ + bx + cf(x)=ax+bx+c, where aaa, bbb, and ccc are constants.3. The function has a maximum or minimum value at its vertex.

STEP 2

The coefficient of x2x^2x2 in the quadratic function determines whether the function has a maximum or minimum value. If the coefficient of x2x^2x2 is positive, the function has a minimum value. If the coefficient of x2x^2x2 is negative, the function has a maximum value.

STEP 3

Identify the coefficient of x2x^2x2 in the given function.f(x)=3x2+18x−9f(x) =3x^2 +18x -9f(x)=3x2+18x−9The coefficient of x2x^2x2 is3.

STEP 4

Since the coefficient of x2x^2x2 is positive, the function has a minimum value.

STEP 5

To find the minimum value, we first need to find the x-coordinate of the vertex. The x-coordinate of the vertex of a quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c is given by −b2a-\frac{b}{2a}−2ab​.

STEP 6

Substitute the values of aaa and bbb into the formula to find the x-coordinate of the vertex.x=−b2a=−182⋅3x = -\frac{b}{2a} = -\frac{18}{2 \cdot3}x=−2ab​=−2⋅318​

STEP 7

Calculate the x-coordinate of the vertex.x=−186=−3x = -\frac{18}{6} = -3x=−618​=−3

STEP 8

Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting this x-coordinate back into the function.f(x)=3(−3)2+18(−3)−f(x) =3(-3)^2 +18(-3) -f(x)=3(−3)2+18(−3)−

STEP 9

Calculate the minimum value of the function.f(−3)=3⋅9−54−9=27−54−9=−36f(-3) =3 \cdot9 -54 -9 =27 -54 -9 = -36f(−3)=3⋅9−54−9=27−54−9=−36The function fff has a minimum value of -36.

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QuestionDetermine if the quadratic function $f(x)=3x^{2}+18x-9$ has a maximum or minimum value and find that value.

Assumptions1. The given function is a quadratic function, which is a polynomial of degree. . The function is in the form $f(x) = ax^ + bx + c$ , where $a$ , $b$ , and $c$ are constants.
3. The function has a maximum or minimum value at its vertex.

The coefficient of $x^2$ in the quadratic function determines whether the function has a maximum or minimum value. If the coefficient of $x^2$ is positive, the function has a minimum value. If the coefficient of $x^2$ is negative, the function has a maximum value.

Identify the coefficient of $x^2$ in the given function.
$f(x) =3x^2 +18x -9$ The coefficient of $x^2$ is3.

Since the coefficient of $x^2$ is positive, the function has a minimum value.

To find the minimum value, we first need to find the x-coordinate of the vertex. The x-coordinate of the vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by $-\frac{b}{2a}$ .

Substitute the values of $a$ and $b$ into the formula to find the x-coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{18}{2 \cdot3}$

Calculate the x-coordinate of the vertex.
$x = -\frac{18}{6} = -3$

Now that we have the x-coordinate of the vertex, we can find the minimum value of the function by substituting this x-coordinate back into the function.
$f(x) =3(-3)^2 +18(-3) -$

Calculate the minimum value of the function.
$f(-3) =3 \cdot9 -54 -9 =27 -54 -9 = -36$ The function $f$ has a minimum value of -36.