is this a function or not a function

Question

y = 8x2

Answer 1

To determine whether the equation \( y = 8x^2 \) represents a function, we can refer to the definition of a function. A relation is considered a function if each input (x-value) has exactly one output (y-value).

In the case of the equation \( y = 8x^2 \):

1. For every \( x \) value you plug into the equation, you will get one and only one corresponding \( y \) value. For example:

   • If \( x = 1 \), then \( y = 8(1)^2 = 8 \).
   • If \( x = -1 \), then \( y = 8(-1)^2 = 8 \) (still one output for two different inputs).

2. The equation is quadratic, which means its graph is a parabola that opens upwards. Each vertical line drawn through the graph will intersect it at most once, which confirms that each input gives only one output.

Thus, \( y = 8x^2 \) is indeed a function.

Answer 2

is this a function?
4x + 53 = y

Answer 3

Yes, the equation \( 4x + 53 = y \) represents a function.

To analyze why it is a function, we can observe the following:

1. Format: The equation can be rearranged to the form \( y = 4x + 53 \). This is the standard form of a linear equation, which is always a function.

2. Single Output: For every value of \( x \) that you input into the equation, you will get exactly one corresponding value of \( y \). This satisfies the definition of a function, which states that each input must be associated with exactly one output.

For example:

• If \( x = 0 \), then \( y = 4(0) + 53 = 53 \).
• If \( x = 1 \), then \( y = 4(1) + 53 = 57 \).

Since there is only one \( y \) output for each value of \( x \), we conclude that \( 4x + 53 = y \) is indeed a function.

Answer 4

A system of equations is given below.
x + 3y = 15
3x + 2y = 10

What is the value of y for the solution to the system? Type your response in the box below.

y =
Math Equation Response Question
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Answer 5

To solve the system of equations:

1. First equation:
   \[ x + 3y = 15 \quad \text{(Equation 1)} \]

2. Second equation:
   \[ 3x + 2y = 10 \quad \text{(Equation 2)} \]

We'll use substitution or elimination. Here, I'll use substitution for clarity.

Step 1: Solve Equation 1 for \( x \)

From Equation 1, we can express \( x \) in terms of \( y \): \[ x = 15 - 3y \]

Step 2: Substitute \( x \) into Equation 2

Now, we substitute \( x \) in Equation 2: \[ 3(15 - 3y) + 2y = 10 \]

Step 3: Simplify and solve for \( y \)

Distributing: \[ 45 - 9y + 2y = 10 \]

Combine like terms: \[ 45 - 7y = 10 \]

Subtract 45 from both sides: \[ -7y = 10 - 45 \] \[ -7y = -35 \]

Divide both sides by -7: \[ y = 5 \]

Answer

The value of \( y \) for the solution to the system is \( \boxed{5} \).

Answer 6

Joe claims that the equation
, where b is a real number, represents the graph of a straight line.

Which statement best classifies Joe's claim?
A.
Joe's claim is true only when b is 0.

B.
Joe's claim is true for all values of b.

C.
Joe's claim is not true for any value of b.

D.
Joe's claim is true only when b is not zero.

Answer 7

To determine Joe's claim about the equation involving \( b \) and whether it represents the graph of a straight line, we need to clarify the equation that Joe is referring to.

However, let’s consider a common form of linear equations. If we assume the equation is like \( y = mx + b \) (the slope-intercept form of a line), where \( m \) and \( b \) are constants, then:

• If \( b \) is any real number, the graph of the equation will always be a straight line, as the slope \( m \) (which may also depend on \( b \)) will determine the angle of the line, and \( b \) (the y-intercept) will determine where the line crosses the y-axis.

Thus, if Joe's claim is about the equation taking the form of \( y = mx + b \), where \( b \) is simply a constant:

• Answer: B. Joe's claim is true for all values of b.

This is because the equation \( y = mx + b \) represents a straight line for any real number \( b \).

If you had a specific equation in mind that you didn't mention, please provide it so we can analyze it properly!

Answer 8

Monique is reading a book. The table shows how the number of pages remaining in the book is changing over time.

READING A BOOK

Time (hours) Number of Pages Remaining
0 360
1 320
2 280
3 240
4 200
Which equation describes the relationship between t, the time in hours, and N, the number of pages remaining in the book?

A. n = -40t - 360

B. n = -40t + 360

C. n = 40t + 360

D. n = 40t - 360

Answer 9

To determine the equation that describes the relationship between \( t \) (time in hours) and \( N \) (the number of pages remaining), we can analyze the data provided in the table.

Analyzing the Data:

Here’s the data from the table:

| Time (hours) | Number of Pages Remaining | |--------------|---------------------------| | 0 | 360 | | 1 | 320 | | 2 | 280 | | 3 | 240 | | 4 | 200 |

Finding the Pattern:

1. Initial Value: At \( t = 0 \) hours, \( N = 360 \) pages are remaining.
2. Change in Pages: Each hour, the number of pages remaining decreases by 40 pages:
   • From 360 to 320 (decrease of 40)
   • From 320 to 280 (decrease of 40)
   • From 280 to 240 (decrease of 40)
   • From 240 to 200 (decrease of 40)

This constant decrease indicates a linear relationship.

Slope and Intercept:

• The slope \( m \) (the rate of change of \( N \) with respect to \( t \)) is -40 (because the number of pages is decreasing).
• The \( y \)-intercept (the number of pages when \( t = 0 \)) is 360.

Formulating the Equation:

The equation of a linear relationship can generally be written in the form: \[ N = mt + b \] where:

• \( m \) is the slope (-40 in this case),
• \( b \) is the y-intercept (360 here).

Putting it all together, we have: \[ N = -40t + 360 \]

Conclusion:

The equation that describes the relationship between \( t \) and \( N \) is:

B. \( N = -40t + 360 \).

Answer 10

Xavior took a total of 124 quarters and dimes to trade in for cash at the bank. He got exactly $25 back. How many quarters did he have?

A.
40

B.
62

C.
84

D.
100

Answer 11

To solve the problem, we need to interpret the information given about Xavior's coins and set up a system of equations.

Let:

• \( q \) be the number of quarters.
• \( d \) be the number of dimes.

From the problem, we have two main pieces of information which can be expressed in equations:

1. Total number of coins: \[ q + d = 124 \quad \text{(Equation 1)} \]

2. Total value of the coins: The value of a quarter is $0.25 and the value of a dime is $0.10. Therefore, the value of all the coins can be expressed as: \[ 0.25q + 0.10d = 25 \quad \text{(Equation 2)} \]

To eliminate the decimals in Equation 2, we can multiply every term by 100: \[ 25q + 10d = 2500 \quad \text{(Equation 2 simplified)} \]

Now we have the system of equations:

1. \( q + d = 124 \)
2. \( 25q + 10d = 2500 \)

Step 1: Solve Equation 1 for \( d \)

From Equation 1: \[ d = 124 - q \]

Step 2: Substitute into Equation 2

Now substitute \( d \) into the simplified Equation 2: \[ 25q + 10(124 - q) = 2500 \]

Step 3: Distribute and Simplify

Distributing the 10: \[ 25q + 1240 - 10q = 2500 \]

Combine like terms: \[ 15q + 1240 = 2500 \]

Step 4: Isolate \( q \)

Subtract 1240 from both sides: \[ 15q = 2500 - 1240 \] \[ 15q = 1260 \]

Now, divide by 15: \[ q = \frac{1260}{15} = 84 \]

Conclusion

So, Xavior had 84 quarters. The answer is:

C. 84.