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Determine if 7x^2 + y = 5 is a linear or non linear function.

a. there is not enough information to determine if 7x^2 + y = 5 is a nonlinear or linear function.
b. the graph of function 7x^2 + y = 5 is not a straight line
c. The function 7x^2 + y = 5 is a non linear.
d. The function 7x^2 + y = 5 is a linear function.

11 answers

To determine whether the equation \( 7x^2 + y = 5 \) is linear or nonlinear, we can rearrange it into slope-intercept form (\( y = mx + b \)) and analyze the relationship between the variables.

Starting with the given equation:

\[ 7x^2 + y = 5 \]

We can isolate \( y \):

\[ y = -7x^2 + 5 \]

In this form, it's clear that the equation describes a quadratic function due to the \( x^2 \) term. A linear function has the general form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, and does not contain any exponent greater than 1.

Since the presence of the \( x^2 \) term indicates that this function is not linear, we can conclude that:

  • The graph of the function \( 7x^2 + y = 5 \) is a parabola, not a straight line.
  • Therefore, it is classified as a nonlinear function.

Given these considerations, the best answer is:

c. The function 7x^2 + y = 5 is a non linear.

Determine the intial value and the rate of change in the following description.

Last year, Joshua's tomato plant reached a maximum height of 2 feet. He is growing another tomato plant this year, and he is tracking its growth. The plant's height today is 10 inches, and it grows 1.7 inches each day.

The initial value is b = ___, and the rate of change is m = ___.

To determine the initial value and the rate of change from the description provided, we can interpret the situation as describing a linear function representing the height of the tomato plant over time.

  1. Initial Value (b): The initial value corresponds to the height of the plant at the start of the measurement. In this case, the plant's height today is 10 inches, and since we are starting the growth tracking from today, the initial height (when \( t = 0 \)) is:

    \[ b = 10 \text{ inches} \]

  2. Rate of Change (m): The rate of change represents how much the height of the plant increases per unit of time, which is daily in this scenario. The problem states that the plant grows 1.7 inches each day, so the rate of change is:

    \[ m = 1.7 \text{ inches per day} \]

Therefore, the answers are:

  • The initial value is \( b = 10 \) inches.
  • The rate of change is \( m = 1.7 \) inches per day.
Use the table to answer the question.

x y
2 22
8 25

Determine the rate of change and the initial value of the linear function given here as a table of values.

The rate of change is m = ___, and the initial value is b = ___.

To find the rate of change \( m \) and the initial value \( b \) for the linear function described by the table of values, we will first calculate the rate of change using the provided points.

  1. Rate of Change (m): The rate of change of a linear function can be calculated using the formula:

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

From the table:

  • Point 1: \( (x_1, y_1) = (2, 22) \)
  • Point 2: \( (x_2, y_2) = (8, 25) \)

Now, substitute the values into the formula:

\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]

So, the rate of change is:

\[ m = \frac{1}{2} \]

  1. Initial Value (b): Since the table provides discrete values for \( x \) and \( y \) and does not explicitly specify a point where \( x = 0 \), we can find the initial value by using the linear equation derived from the slope-intercept form \( y = mx + b \).

Using one of the points to find \( b \), let's take the point \( (2, 22) \):

\[ y = mx + b \] \[ 22 = \frac{1}{2} \cdot 2 + b \] \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]

Therefore, the initial value is:

\[ b = 21 \]

In summary:

  • The rate of change is \( m = \frac{1}{2} \).
  • The initial value is \( b = 21 \).
Use the image to answer the question.
(-3, 4) (0, 3) (3,2) (6, 1)

Determine the initial values and the rate of change of the linear function as given in the graph. Round the answer to three decimal places as needed.

The initial value is ___, and the rate of change is ___.

To analyze the linear function represented by the given points \((-3, 4)\), \((0, 3)\), \((3, 2)\), and \((6, 1)\), we will determine the initial value and the rate of change.

Step 1: Identify the Initial Value (b)

The initial value in the context of a linear function is the value of \(y\) when \(x = 0\). From the points provided, the point that corresponds to \(x = 0\) is:

\[ (0, 3) \]

Thus, the initial value \(b\) is:

\[ b = 3 \]

Step 2: Calculate the Rate of Change (m)

The rate of change of a linear function can be determined using the slope formula:

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

We can use any two points; let's use the points \((0, 3)\) and \((3, 2)\):

  • \( (x_1, y_1) = (0, 3) \)
  • \( (x_2, y_2) = (3, 2) \)

Substituting into the formula:

\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \approx -0.333 \]

Thus, the rate of change \(m\) rounded to three decimal places is:

\[ m \approx -0.333 \]

Summary

  • The initial value is \(3\).
  • The rate of change is approximately \(-0.333\).

So the answers are:

  • The initial value is \(3\).
  • The rate of change is \(-0.333\).
A pan with four eggs weights 18 ounces. A pan with eight eggs weighs 24 ounces. What is the weight of the pan?

Let's denote the weight of the pan as \( p \) ounces and the weight of each egg as \( e \) ounces.

From the problem, we have two equations based on the information provided:

  1. A pan with four eggs weighs 18 ounces: \[ p + 4e = 18 \quad \text{(1)} \]

  2. A pan with eight eggs weighs 24 ounces: \[ p + 8e = 24 \quad \text{(2)} \]

Now, we can solve these two equations to find \( p \) and \( e \).

Step 1: Solve for \( p \) in terms of \( e \) from equation (1)

From equation (1): \[ p = 18 - 4e \quad \text{(3)} \]

Step 2: Substitute equation (3) into equation (2)

Substituting (3) into (2): \[ (18 - 4e) + 8e = 24 \]

Step 3: Simplify the equation

Combine like terms: \[ 18 + 4e = 24 \]

Step 4: Solve for \( e \)

Subtract 18 from both sides: \[ 4e = 24 - 18 \] \[ 4e = 6 \] \[ e = \frac{6}{4} = 1.5 \quad \text{(the weight of each egg in ounces)} \]

Step 5: Substitute back to find \( p \)

Substitute \( e = 1.5 \) back into equation (3): \[ p = 18 - 4(1.5) \] \[ p = 18 - 6 = 12 \]

Conclusion

The weight of the pan is \( \boxed{12} \) ounces.

Manuel got a new job and was given a hiring bonus along with his monthly salary. His earring after x months are modeled by the function.
f (x) = 1,500x + 300. How much did he receive as a hiring bonus?

The function \( f(x) = 1,500x + 300 \) models Manuel's earnings after \( x \) months, where:

  • \( 1,500x \) represents his monthly salary (which is $1,500 per month).
  • The constant term \( 300 \) represents the hiring bonus he received when he was hired.

Since the hiring bonus is the constant part of the equation, we can conclude that:

Manuel received a hiring bonus of \( \boxed{300} \) dollars.