Question1. Find $f(-2)$ where $f(x)=-2x^2-4^x$ .
2. Graph $y=\frac{1}{2}(x-4)^2+2$ and convert to standard form.
3. Factor $x^2-8x+15$ and $2x^2+7x-15$ .
4. Solve $2x^2+8x-15=0$ using the quadratic formula.
5. Find the function, in factored form, for a parabola with $x$ -intercepts of -5 and 7, passing through $(6,33)$ .

Studdy Solution

STEP 1

Assumptions
1. For problem 2, the function $f(x)$ is given as $f(x)=-2x^{2}-4^{x}$ .
2. For problem 3, we are asked to graph the function $y=\frac{1}{2}(x-4)^{2}+2$ and convert it to standard form.
3. For problem 4, we need to factor two quadratic expressions: $x^{2}-8x+15$ and $2x^{2}+7x-15$ .
4. For problem 5, we are solving the quadratic equation $2x^{2}+8x-15=0$ using the quadratic formula or by completing the square.
5. For problem 7, we are finding the equation of a parabola in factored form with x-intercepts of -5 and 7, and passing through the point $(6,33)$ .

STEP 2

To determine the value of $f(-2)$ for the function $f(x)=-2x^{2}-4^{x}$ , we substitute $x$ with $-2$ .
$f(-2) = -2(-2)^{2}-4^{-2}$

STEP 3

Evaluate the exponent $(-2)^{2}$ .
$(-2)^{2} = 4$

STEP 4

Substitute the value from STEP_3 into the expression.
$f(-2) = -2(4)-4^{-2}$

STEP 5

Evaluate the negative exponent $4^{-2}$ .
$4^{-2} = \frac{1}{4^{2}} = \frac{1}{16}$

STEP 6

Substitute the value from STEP_5 into the expression.
$f(-2) = -2(4)-\frac{1}{16}$

STEP 7

Multiply $-2$ by $4$ .
$f(-2) = -8 - \frac{1}{16}$

STEP 8

Combine the terms to find the value of $f(-2)$ .
$f(-2) = -8 - \frac{1}{16} = -\frac{128}{16} - \frac{1}{16} = -\frac{129}{16}$
The value of $f(-2)$ is $-\frac{129}{16}$ .

STEP 9

To graph the function $y=\frac{1}{2}(x-4)^{2}+2$ , we recognize that it is a parabola in vertex form, where the vertex is $(h, k)$ with $h=4$ and $k=2$ .

STEP 10

Plot the vertex $(4, 2)$ on the graph.

STEP 11

Since the coefficient of $(x-4)^{2}$ is $\frac{1}{2}$ , which is positive, the parabola opens upwards.

STEP 12

Find additional points by choosing $x$ -values around the vertex and calculating the corresponding $y$ -values.

STEP 13

Plot these points and draw a smooth curve through them to complete the graph of the parabola.

STEP 14

To convert the function to standard form, we need to expand the squared term and combine like terms.
$y = \frac{1}{2}(x-4)^{2}+2$

STEP 15

Expand the squared term.
$(x-4)^{2} = x^{2} - 8x + 16$

STEP 16

Multiply the expanded term by $\frac{1}{2}$ .
$\frac{1}{2}(x^{2} - 8x + 16) = \frac{1}{2}x^{2} - 4x + 8$

STEP 17

Combine the term from STEP_16 with the constant term $+2$ .
$y = \frac{1}{2}x^{2} - 4x + 8 + 2$

STEP 18

Combine the constant terms.
$y = \frac{1}{2}x^{2} - 4x + 10$
The standard form of the function is $y = \frac{1}{2}x^{2} - 4x + 10$ .

STEP 19

To factor the quadratic expression $x^{2}-8x+15$ , we look for two numbers that multiply to $15$ and add to $-8$ .

STEP 20

The numbers that satisfy these conditions are $-3$ and $-5$ .

STEP 21

Write the factored form using these numbers.
$x^{2}-8x+15 = (x-3)(x-5)$
The factored form of $x^{2}-8x+15$ is $(x-3)(x-5)$ .

STEP 22

To factor the quadratic expression $2x^{2}+7x-15$ , we look for two numbers that multiply to $2 \times (-15) = -30$ and add to $7$ .

STEP 23

The numbers that satisfy these conditions are $10$ and $-3$ .

STEP 24

Write the factored form using these numbers.
$2x^{2}+7x-15 = (2x-3)(x+5)$
The factored form of $2x^{2}+7x-15$ is $(2x-3)(x+5)$ .

STEP 25

To solve the quadratic equation $2x^{2}+8x-15=0$ , we use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

STEP 26

Identify the coefficients $a=2$ , $b=8$ , and $c=-15$ .

STEP 27

Substitute these values into the quadratic formula.
$x = \frac{-8 \pm \sqrt{8^{2}-4(2)(-15)}}{2(2)}$

STEP 28

Calculate the discriminant $\sqrt{8^{2}-4(2)(-15)}$ .
$\sqrt{64 + 120} = \sqrt{184}$

STEP 29

Simplify the expression under the square root to get the exact value or use a calculator to find the approximate value.
$x = \frac{-8 \pm \sqrt{184}}{4}$

STEP 30

Round the answers to two decimal places if necessary.
$x \approx \frac{-8 \pm 13.56}{4}$
$x_1 \approx \frac{5.56}{4} \approx 1.39$
$x_2 \approx \frac{-21.56}{4} \approx -5.39$
The solutions to the quadratic equation are approximately $x_1 \approx 1.39$ and $x_2 \approx -5.39$ .

STEP 31

To determine the function of a parabola with x-intercepts of -5 and 7, and passing through the point $(6,33)$ , we start with the factored form.
$y = a(x + 5)(x - 7)$

STEP 32

Substitute the point $(6,33)$ into the equation to solve for $a$ .
$33 = a(6 + 5)(6 - 7)$

STEP 33

Simplify the equation.
$33 = a(11)(-1)$

STEP 34

Solve for $a$ .
$a = -\frac{33}{11} = -3$

STEP 35

Write the final equation of the parabola using the value of $a$ .
$y = -3(x + 5)(x - 7)$
The function of the parabola in factored form is $y = -3(x + 5)(x - 7)$ .

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Studdy Solution

STEP 1

STEP 2

To determine the value of f(−2)f(-2)f(−2) for the function f(x)=−2x2−4xf(x)=-2x^{2}-4^{x}f(x)=−2x2−4x, we substitute xxx with −2-2−2.f(−2)=−2(−2)2−4−2 f(-2) = -2(-2)^{2}-4^{-2} f(−2)=−2(−2)2−4−2

STEP 3

Evaluate the exponent (−2)2(-2)^{2}(−2)2.(−2)2=4 (-2)^{2} = 4 (−2)2=4

STEP 4

Substitute the value from STEP_3 into the expression.f(−2)=−2(4)−4−2 f(-2) = -2(4)-4^{-2} f(−2)=−2(4)−4−2

STEP 5

Evaluate the negative exponent 4−24^{-2}4−2.4−2=142=116 4^{-2} = \frac{1}{4^{2}} = \frac{1}{16} 4−2=421​=161​

STEP 6

Substitute the value from STEP_5 into the expression.f(−2)=−2(4)−116 f(-2) = -2(4)-\frac{1}{16} f(−2)=−2(4)−161​

STEP 7

Multiply −2-2−2 by 444.f(−2)=−8−116 f(-2) = -8 - \frac{1}{16} f(−2)=−8−161​

STEP 8

STEP 9

To graph the function y=12(x−4)2+2y=\frac{1}{2}(x-4)^{2}+2y=21​(x−4)2+2, we recognize that it is a parabola in vertex form, where the vertex is (h,k)(h, k)(h,k) with h=4h=4h=4 and k=2k=2k=2.

STEP 10

Plot the vertex (4,2)(4, 2)(4,2) on the graph.

STEP 11

Since the coefficient of (x−4)2(x-4)^{2}(x−4)2 is 12\frac{1}{2}21​, which is positive, the parabola opens upwards.

STEP 12

Find additional points by choosing xxx-values around the vertex and calculating the corresponding yyy-values.

STEP 13

Plot these points and draw a smooth curve through them to complete the graph of the parabola.

STEP 14

To convert the function to standard form, we need to expand the squared term and combine like terms.y=12(x−4)2+2 y = \frac{1}{2}(x-4)^{2}+2 y=21​(x−4)2+2

STEP 15

Expand the squared term.(x−4)2=x2−8x+16 (x-4)^{2} = x^{2} - 8x + 16 (x−4)2=x2−8x+16

STEP 16

Multiply the expanded term by 12\frac{1}{2}21​.12(x2−8x+16)=12x2−4x+8 \frac{1}{2}(x^{2} - 8x + 16) = \frac{1}{2}x^{2} - 4x + 8 21​(x2−8x+16)=21​x2−4x+8

STEP 17

Combine the term from STEP_16 with the constant term +2+2+2.y=12x2−4x+8+2 y = \frac{1}{2}x^{2} - 4x + 8 + 2 y=21​x2−4x+8+2

STEP 18

Combine the constant terms.y=12x2−4x+10 y = \frac{1}{2}x^{2} - 4x + 10 y=21​x2−4x+10The standard form of the function is y=12x2−4x+10y = \frac{1}{2}x^{2} - 4x + 10y=21​x2−4x+10.

STEP 19

To factor the quadratic expression x2−8x+15x^{2}-8x+15x2−8x+15, we look for two numbers that multiply to 151515 and add to −8-8−8.

STEP 20

The numbers that satisfy these conditions are −3-3−3 and −5-5−5.

STEP 21

Write the factored form using these numbers.x2−8x+15=(x−3)(x−5) x^{2}-8x+15 = (x-3)(x-5) x2−8x+15=(x−3)(x−5)The factored form of x2−8x+15x^{2}-8x+15x2−8x+15 is (x−3)(x−5)(x-3)(x-5)(x−3)(x−5).

STEP 22

To factor the quadratic expression 2x2+7x−152x^{2}+7x-152x2+7x−15, we look for two numbers that multiply to 2×(−15)=−302 \times (-15) = -302×(−15)=−30 and add to 777.

STEP 23

The numbers that satisfy these conditions are 101010 and −3-3−3.

STEP 24

Write the factored form using these numbers.2x2+7x−15=(2x−3)(x+5) 2x^{2}+7x-15 = (2x-3)(x+5) 2x2+7x−15=(2x−3)(x+5)The factored form of 2x2+7x−152x^{2}+7x-152x2+7x−15 is (2x−3)(x+5)(2x-3)(x+5)(2x−3)(x+5).

STEP 25

To solve the quadratic equation 2x2+8x−15=02x^{2}+8x-15=02x2+8x−15=0, we use the quadratic formula:x=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} x=2a−b±b2−4ac​​

STEP 26

Identify the coefficients a=2a=2a=2, b=8b=8b=8, and c=−15c=-15c=−15.

STEP 27

Substitute these values into the quadratic formula.x=−8±82−4(2)(−15)2(2) x = \frac{-8 \pm \sqrt{8^{2}-4(2)(-15)}}{2(2)} x=2(2)−8±82−4(2)(−15)​​

STEP 28

Calculate the discriminant 82−4(2)(−15)\sqrt{8^{2}-4(2)(-15)}82−4(2)(−15)​.64+120=184 \sqrt{64 + 120} = \sqrt{184} 64+120​=184​

STEP 29

Simplify the expression under the square root to get the exact value or use a calculator to find the approximate value.x=−8±1844 x = \frac{-8 \pm \sqrt{184}}{4} x=4−8±184​​

STEP 30

STEP 31

To determine the function of a parabola with x-intercepts of -5 and 7, and passing through the point (6,33)(6,33)(6,33), we start with the factored form.y=a(x+5)(x−7) y = a(x + 5)(x - 7) y=a(x+5)(x−7)

STEP 32

Substitute the point (6,33)(6,33)(6,33) into the equation to solve for aaa.33=a(6+5)(6−7) 33 = a(6 + 5)(6 - 7) 33=a(6+5)(6−7)

STEP 33

Simplify the equation.33=a(11)(−1) 33 = a(11)(-1) 33=a(11)(−1)

STEP 34

Solve for aaa.a=−3311=−3 a = -\frac{33}{11} = -3 a=−1133​=−3

STEP 35

Write the final equation of the parabola using the value of aaa.y=−3(x+5)(x−7) y = -3(x + 5)(x - 7) y=−3(x+5)(x−7)The function of the parabola in factored form is y=−3(x+5)(x−7)y = -3(x + 5)(x - 7)y=−3(x+5)(x−7).

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To determine the value of $f(-2)$ for the function $f(x)=-2x^{2}-4^{x}$ , we substitute $x$ with $-2$ .
$f(-2) = -2(-2)^{2}-4^{-2}$

Evaluate the exponent $(-2)^{2}$ .
$(-2)^{2} = 4$

Substitute the value from STEP_3 into the expression.
$f(-2) = -2(4)-4^{-2}$

Evaluate the negative exponent $4^{-2}$ .
$4^{-2} = \frac{1}{4^{2}} = \frac{1}{16}$

Substitute the value from STEP_5 into the expression.
$f(-2) = -2(4)-\frac{1}{16}$

Multiply $-2$ by $4$ .
$f(-2) = -8 - \frac{1}{16}$

To graph the function $y=\frac{1}{2}(x-4)^{2}+2$ , we recognize that it is a parabola in vertex form, where the vertex is $(h, k)$ with $h=4$ and $k=2$ .

Plot the vertex $(4, 2)$ on the graph.

Since the coefficient of $(x-4)^{2}$ is $\frac{1}{2}$ , which is positive, the parabola opens upwards.

Find additional points by choosing $x$ -values around the vertex and calculating the corresponding $y$ -values.

To convert the function to standard form, we need to expand the squared term and combine like terms.
$y = \frac{1}{2}(x-4)^{2}+2$

Expand the squared term.
$(x-4)^{2} = x^{2} - 8x + 16$

Multiply the expanded term by $\frac{1}{2}$ .
$\frac{1}{2}(x^{2} - 8x + 16) = \frac{1}{2}x^{2} - 4x + 8$

Combine the term from STEP_16 with the constant term $+2$ .
$y = \frac{1}{2}x^{2} - 4x + 8 + 2$

Combine the constant terms.
$y = \frac{1}{2}x^{2} - 4x + 10$
The standard form of the function is $y = \frac{1}{2}x^{2} - 4x + 10$ .

To factor the quadratic expression $x^{2}-8x+15$ , we look for two numbers that multiply to $15$ and add to $-8$ .

The numbers that satisfy these conditions are $-3$ and $-5$ .

Write the factored form using these numbers.
$x^{2}-8x+15 = (x-3)(x-5)$
The factored form of $x^{2}-8x+15$ is $(x-3)(x-5)$ .

To factor the quadratic expression $2x^{2}+7x-15$ , we look for two numbers that multiply to $2 \times (-15) = -30$ and add to $7$ .

The numbers that satisfy these conditions are $10$ and $-3$ .

Write the factored form using these numbers.
$2x^{2}+7x-15 = (2x-3)(x+5)$
The factored form of $2x^{2}+7x-15$ is $(2x-3)(x+5)$ .

To solve the quadratic equation $2x^{2}+8x-15=0$ , we use the quadratic formula:
$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

Identify the coefficients $a=2$ , $b=8$ , and $c=-15$ .

Substitute these values into the quadratic formula.
$x = \frac{-8 \pm \sqrt{8^{2}-4(2)(-15)}}{2(2)}$

Calculate the discriminant $\sqrt{8^{2}-4(2)(-15)}$ .
$\sqrt{64 + 120} = \sqrt{184}$

Simplify the expression under the square root to get the exact value or use a calculator to find the approximate value.
$x = \frac{-8 \pm \sqrt{184}}{4}$

To determine the function of a parabola with x-intercepts of -5 and 7, and passing through the point $(6,33)$ , we start with the factored form.
$y = a(x + 5)(x - 7)$

Substitute the point $(6,33)$ into the equation to solve for $a$ .
$33 = a(6 + 5)(6 - 7)$

Simplify the equation.
$33 = a(11)(-1)$

Solve for $a$ .
$a = -\frac{33}{11} = -3$

Write the final equation of the parabola using the value of $a$ .
$y = -3(x + 5)(x - 7)$
The function of the parabola in factored form is $y = -3(x + 5)(x - 7)$ .