Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an left parenthesis, x, comma, y, right parenthesis(x,y) point.

Question

y, equals, minus, 3, x, squared, plus, 36, x, minus, 120
y=−3x
2
+36x−120

Answer 1

To find the vertex of the parabola given by the equation \( y = -3x^2 + 36x - 120 \), we can use the formula for the x-coordinate of the vertex of a parabola in the form \( y = ax^2 + bx + c \), which is given by:

\[ x = -\frac{b}{2a} \]

Here, \( a = -3 \) and \( b = 36 \).

Plugging in the values:

\[ x = -\frac{36}{2(-3)} = -\frac{36}{-6} = 6 \]

Now that we have the x-coordinate, we can find the y-coordinate by substituting \( x = 6 \) back into the original equation:

\[ y = -3(6)^2 + 36(6) - 120 \] \[ y = -3(36) + 216 - 120 \] \[ y = -108 + 216 - 120 \] \[ y = 108 - 120 = -12 \]

Thus, the coordinates of the vertex of the parabola are:

\[ \boxed{(6, -12)} \]

Answer 2

Determine which of the following ordered pairs is a solution to the equation 3x−8=−4y
.(1 point)
Responses

(−1,1)
left parenthesis negative 1 comma 1 right parenthesis

(0,2)
left parenthesis 0 comma 2 right parenthesis

(1,1)
left parenthesis 1 comma 1 right parenthesis

(0,−2)

Answer 3

To determine which of the given ordered pairs is a solution to the equation \( 3x - 8 = -4y \), we'll substitute each pair into the equation and see if it holds true.

1. For the pair \( (-1, 1) \): \[ 3(-1) - 8 = -4(1) \] \[ -3 - 8 = -4 \] \[ -11 \neq -4 \quad \text{(Not a solution)} \]

2. For the pair \( (0, 2) \): \[ 3(0) - 8 = -4(2) \] \[ -8 = -8 \quad \text{(Solution)} \]

3. For the pair \( (1, 1) \): \[ 3(1) - 8 = -4(1) \] \[ 3 - 8 = -4 \] \[ -5 \neq -4 \quad \text{(Not a solution)} \]

4. For the pair \( (0, -2) \): \[ 3(0) - 8 = -4(-2) \] \[ -8 = 8 \quad \text{(Not a solution)} \]

The only ordered pair that satisfies the equation \( 3x - 8 = -4y \) is:

\[ (0, 2) \]

Thus, the correct answer is \( (0, 2) \).

Answer 4

The math teacher asked his students to determine a solution for 1.5x−2y=8
. Which solution below is the correct one?(1 point)
Responses

Sam believes (−4,−7)
is a solution.
Sam believes left parenthesis negative 4 comma negative 7 right parenthesis is a solution.

Mary believes (4,−7)
is a solution.
Mary believes left parenthesis 4 comma negative 7 right parenthesis is a solution.

Fiona believes (4,7)
is a solution.
Fiona believes left parenthesis 4 comma 7 right parenthesis is a solution.

Peter believes (−4,7)
is a solution.

Answer 5

To determine which of the given ordered pairs is a solution to the equation \( 1.5x - 2y = 8 \), we will substitute each pair into the equation and check if it holds true.

1. For Sam's pair \( (-4, -7) \): \[ 1.5(-4) - 2(-7) = 8 \] \[ -6 + 14 = 8 \] \[ 8 = 8 \quad \text{(This is a solution.)} \]

2. For Mary's pair \( (4, -7) \): \[ 1.5(4) - 2(-7) = 8 \] \[ 6 + 14 = 8 \] \[ 20 \neq 8 \quad \text{(Not a solution)} \]

3. For Fiona's pair \( (4, 7) \): \[ 1.5(4) - 2(7) = 8 \] \[ 6 - 14 = 8 \] \[ -8 \neq 8 \quad \text{(Not a solution)} \]

4. For Peter's pair \( (-4, 7) \): \[ 1.5(-4) - 2(7) = 8 \] \[ -6 - 14 = 8 \] \[ -20 \neq 8 \quad \text{(Not a solution)} \]

Thus, the only correct solution is from Sam:

\[ (-4, -7) \]

So the answer is \( \text{Sam believes } (-4, -7) \text{ is a solution.} \)

Answer 6

Select the option below that correctly shows which of the following ordered pairs is part of the solution set for the equation y=4x+2
.(1 point)
Responses

(3,7)
left parenthesis 3 comma 7 right parenthesis

(3,45)
left parenthesis 3 comma 45 right parenthesis

(3,14)
left parenthesis 3 comma 14 right parenthesis

(3,12)

Answer 7

To determine which of the given ordered pairs is part of the solution set for the equation \( y = 4x + 2 \), we will substitute \( x \) from each pair into the equation and check if the resulting \( y \) matches the \( y \) in the ordered pairs.

1. For the pair \( (3, 7) \): \[ y = 4(3) + 2 = 12 + 2 = 14 \] \[ 7 \neq 14 \quad \text{(Not a solution)} \]

2. For the pair \( (3, 45) \): \[ y = 4(3) + 2 = 12 + 2 = 14 \] \[ 45 \neq 14 \quad \text{(Not a solution)} \]

3. For the pair \( (3, 14) \): \[ y = 4(3) + 2 = 12 + 2 = 14 \] \[ 14 = 14 \quad \text{(This is a solution.)} \]

4. For the pair \( (3, 12) \): \[ y = 4(3) + 2 = 12 + 2 = 14 \] \[ 12 \neq 14 \quad \text{(Not a solution)} \]

The only ordered pair that is part of the solution set for the equation \( y = 4x + 2 \) is:

\[ (3, 14) \]

Thus, the correct option is \( (3, 14) \).

Answer 8

Use the image to answer the question.

A graph shows both the x-axis and y-axis, ranging from negative 10 to 10 in increments of 5. A line passes through the first, second, and third quadrants.

Which of the following ordered pairs is part of the solution set for the equation y=2x+5
?

(1 point)
Responses

(1,2)
left parenthesis 1 comma 2 right parenthesis

(1,7)
left parenthesis 1 comma 7 right parenthesis

(3,−1)
left parenthesis 3 comma negative 1 right parenthesis

(7,1)
left parenthesis 7 comma 1 right parenthesis
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Answer 9

To determine which of the given ordered pairs is part of the solution set for the equation \( y = 2x + 5 \), we will substitute \( x \) from each pair into the equation and check if the resulting \( y \) matches the \( y \) in the ordered pairs.

1. For the pair \( (1, 2) \): \[ y = 2(1) + 5 = 2 + 5 = 7 \] \[ 2 \neq 7 \quad \text{(Not a solution)} \]

2. For the pair \( (1, 7) \): \[ y = 2(1) + 5 = 2 + 5 = 7 \] \[ 7 = 7 \quad \text{(This is a solution.)} \]

3. For the pair \( (3, -1) \): \[ y = 2(3) + 5 = 6 + 5 = 11 \] \[ -1 \neq 11 \quad \text{(Not a solution)} \]

4. For the pair \( (7, 1) \): \[ y = 2(7) + 5 = 14 + 5 = 19 \] \[ 1 \neq 19 \quad \text{(Not a solution)} \]

The only ordered pair that is part of the solution set for the equation \( y = 2x + 5 \) is:

\[ (1, 7) \]

So the correct option is \( (1, 7) \).

Answer 10

Use the image to answer the question.

A graph shows both the x-axis and y-axis, ranging from negative 10 to 10 in increments of 5. An upward parabola passes through all four quadrants.

Which of the following ordered pairs is part of the solution set for the equation y=5x2−2
?

(1 point)
Responses

(−1,5)
left parenthesis negative 1 comma 5 right parenthesis

(−1,3)
left parenthesis negative 1 comma 3 right parenthesis

(1,5)
left parenthesis 1 comma 5 right parenthesis

(−1,43)

Answer 11

To determine which of the given ordered pairs is part of the solution set for the equation \( y = 5x^2 - 2 \), we will substitute \( x \) values from each pair into the equation and check if the resulting \( y \) matches the \( y \) in the ordered pairs.

1. For the pair \( (-1, 5) \): \[ y = 5(-1)^2 - 2 = 5(1) - 2 = 5 - 2 = 3 \] \[ 5 \neq 3 \quad \text{(Not a solution)} \]

2. For the pair \( (-1, 3) \): \[ y = 5(-1)^2 - 2 = 5(1) - 2 = 5 - 2 = 3 \] \[ 3 = 3 \quad \text{(This is a solution.)} \]

3. For the pair \( (1, 5) \): \[ y = 5(1)^2 - 2 = 5(1) - 2 = 5 - 2 = 3 \] \[ 5 \neq 3 \quad \text{(Not a solution)} \]

4. For the pair \( (-1, 43) \): \[ y = 5(-1)^2 - 2 = 5(1) - 2 = 5 - 2 = 3 \] \[ 43 \neq 3 \quad \text{(Not a solution)} \]

The only ordered pair that satisfies the equation \( y = 5x^2 - 2 \) is:

\[ (-1, 3) \]

So the correct option is \( (-1, 3) \).