interpret the equation

Question

y = -4x + 10. What type of function does this equation define? Describe its graph.

a. this is a nonlinear function. Its graph is a curve.
b. This is a linear function. its graph is a straight line with a slope of -4 and a y - intercept of 10.
c. This is a linear function. Its graph is straight line with a y-intercept of -4 and a slope of 10.
d. This is a nonlinear function. Its graph has a maximum at (0, 10).

Answer 1

The equation \( y = -4x + 10 \) defines a linear function. Here's the breakdown:

• The equation is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• In this case, the slope \( m = -4 \) indicates that for every unit increase in \( x \), \( y \) decreases by 4 units (which means the line is descending).
• The y-intercept \( b = 10 \) means that the line crosses the y-axis at the point (0, 10).

So, the correct interpretation of the equation is:

b. This is a linear function. Its graph is a straight line with a slope of -4 and a y-intercept of 10.

Answer 2

The equation of a function is y = 2x - 6. If this function were graphed, what would its y-intercept be?

a. 2
b. 6
c. -3
d. -6

Answer 3

To find the y-intercept of the function \( y = 2x - 6 \), we can look for the value of \( y \) when \( x = 0 \).

Substituting \( x = 0 \) into the equation:

\[ y = 2(0) - 6 = 0 - 6 = -6 \]

Thus, the y-intercept of the function is \(-6\).

The correct answer is:

d. -6

Answer 4

Use the table to answer the question.
x y
4 38
6 58

Write the equation of the linear function that models the relationship shown in the table.

a. y = 10x - 2
b. y = 10x + 2
c. y = -10x + 78
d. y = x + 34

Answer 5

To find the equation of the linear function, we can use the points given in the table. The points are (4, 38) and (6, 58).

Step 1: Calculate the Slope \( m \)

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points (4, 38) and (6, 58) where:

• \( (x_1, y_1) = (4, 38) \)
• \( (x_2, y_2) = (6, 58) \)

Calculating the slope:

\[ m = \frac{58 - 38}{6 - 4} = \frac{20}{2} = 10 \]

Step 2: Use the Point-Slope Form to Find the Equation

We can use the point-slope form of a linear equation which is:

\[ y - y_1 = m(x - x_1) \]

Using the slope \( m = 10 \) and the point (4, 38):

\[ y - 38 = 10(x - 4) \]

Step 3: Simplifying the Equation

Now we can simplify it:

1. Distribute \( 10 \):

   \[ y - 38 = 10x - 40 \]

2. Add 38 to both sides:

   \[ y = 10x - 40 + 38 \]

   \[ y = 10x - 2 \]

Conclusion

The equation of the linear function that models the relationship shown in the table is:

a. \( y = 10x - 2 \).

Answer 6

Oliwia is opening a bank account. She learns that thee is a special promotion where if she deposits $3,000 and then makes no additional deposits or withdrawals, her account will earn $3 in interest per month. Write the equation of a linear function that models the relationship between number of months and total amount of money Oliwia will have in her account.

a. y = 3x + 3,000, where x is the number of months and y is the total amount of money Oliwia will have in her account.

b. y = 1,000x + 3, where x is the number of months and y is the total amount of money Oliwia will have in her account.

c. y = 3,000x + 3, where x is the number of months and y is the total amount of money Oliwia will have in her account.

d. y = 3x + 1,000, where x is the number of months and y is the total amount of money Oliwia will have in her account.

Answer 7

To model the relationship between the number of months (\(x\)) and the total amount of money Oliwia will have in her account (\(y\)), we start with the information given:

• Oliwia makes an initial deposit of $3,000.
• She earns $3 in interest per month.

Step 1: Setting Up the Equation

The total amount of money in her account after \(x\) months can be modeled as:

\[ y = \text{initial deposit} + \text{interest per month} \times \text{number of months} \]

Substituting the values:

\[ y = 3000 + 3x \]

Step 2: Rearranging the Equation

This can also be expressed as:

\[ y = 3x + 3000 \]

Conclusion

Therefore, the correct equation of a linear function that models the relationship is:

a. \( y = 3x + 3,000 \), where \( x \) is the number of months and \( y \) is the total amount of money Oliwia will have in her account.

Answer 8

Use the graph to answer the question.
(-1,4)
(1,0)

Write the equation of the linear function shown on the graph.

a. y = -2x - 2
b. y = -2x + 2
c. y = 2x - 2
d. y = 2x + 2

Answer 9

To find the equation of the linear function using the two points given, \((-1, 4)\) and \((1, 0)\), we will follow these steps:

Step 1: Calculate the Slope \(m\)

The slope \(m\) is calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's assign:

• \((x_1, y_1) = (-1, 4)\)
• \((x_2, y_2) = (1, 0)\)

Now plug in the values:

\[ m = \frac{0 - 4}{1 - (-1)} = \frac{-4}{1 + 1} = \frac{-4}{2} = -2 \]

Step 2: Use the Point-Slope Form

We will use the point-slope form of a linear equation, which is:

\[ y - y_1 = m(x - x_1) \]

We can use point \((1, 0)\) and the slope \(m = -2\):

\[ y - 0 = -2(x - 1) \]

Simplifying this:

1. Distributing the slope:

\[ y = -2(x - 1) \implies y = -2x + 2 \]

Conclusion

Thus, the equation of the linear function is:

b. \( y = -2x + 2 \).