Sarah is saving money for a used car that cost $4500. She is able to save $300 every week from her paycheck. The function that models the money saved is m(w)=300w. The graph of the function is shown

Answer 1

To analyze Sarah's savings situation, we can look at the information provided:

1. Cost of the Car: $4500
2. Weekly Savings: $300
3. Function for Savings: \( m(w) = 300w \) where \( w \) is the number of weeks.

Key Points

• Initial Savings: At \( w = 0 \) (when she just starts saving), \( m(0) = 300 \times 0 = 0 \).
• Weekly Savings: For every week that Sarah saves, she increases her savings by $300.
  • For example:
    • After 1 week: \( m(1) = 300 \times 1 = 300 \)
    • After 2 weeks: \( m(2) = 300 \times 2 = 600 \)
    • After 3 weeks: \( m(3) = 300 \times 3 = 900 \)
• Finding the Number of Weeks to Reach $4500: To find out how many weeks it will take to save $4500, we set the savings function equal to the cost of the car: \[ 300w = 4500 \] Dividing both sides by 300: \[ w = \frac{4500}{300} = 15 \] So, it will take Sarah 15 weeks to save enough money for the car.

Graph Representation

On a graph where the x-axis represents time in weeks (\( w \)) and the y-axis represents money saved (\( m(w) \)):

• The line representing the function \( m(w) = 300w \) will have:
  • A y-intercept at (0, 0)
  • A slope of 300, meaning for every 1 week, the amount saved increases by $300.
• The point where the line intersects the line representing $4500 will be at \( (15, 4500) \).

Conclusion

Sarah will have saved enough money to buy her used car after 15 weeks. If you have further questions or need assistance with specific calculations or graphing, feel free to ask!

Answer 2

y=x+2−−−−−√3
is a(n)
square root
function.

Answer 3

The expression you provided, \( y = \sqrt{x + 2} - 3 \), represents a square root function. Let's analyze its components:

1. Square Root Function: A function is classified as a square root function if it contains the square root of a variable expression. In this function, \( \sqrt{x + 2} \) indicates the presence of the square root of \( x + 2 \).

2. Transformation: The function can be interpreted as a transformation of the parent square root function \( y = \sqrt{x} \):

   • The expression \( x + 2 \) shifts the graph to the left by 2 units.
   • The \( -3 \) at the end shifts the graph down by 3 units.

Summary

Thus, \( y = \sqrt{x + 2} - 3 \) is indeed a square root function, and it reflects transformations of the basic square root shape. If you have further questions or need additional clarifications, feel free to ask!

Answer 4

y=x3+3
is a(n)
Cube root
function.

Answer 5

The expression you provided, \( y = x^3 + 3 \), is actually a cubic function, not a cube root function. Let's clarify the distinction:

1. Cubic Function: A function of the form \( y = ax^3 + bx^2 + cx + d \) is classified as a cubic function. In your case, \( y = x^3 + 3 \) can be seen as a cubic function where:

   • \( a = 1 \) (the coefficient of \( x^3 \)),
   • \( b = 0 \) (no \( x^2 \) term),
   • \( c = 0 \) (no \( x \) term),
   • \( d = 3 \) (the constant term).

2. Transformations: The function can be interpreted as a transformation of the parent cubic function \( y = x^3 \):

   • The addition of 3 shifts the graph of the cubic function vertically upward by 3 units.

Summary

• Cubic Function: \( y = x^3 + 3 \) is a cubic function.
• Cube Root Function: A cube root function would have the form \( y = \sqrt[3]{x} \) or some transformation of it.

If you have more questions or need further clarification, feel free to ask!

Answer 6

y=x+2
is a(n)
linear
function.

Answer 7

Yes, you are correct! The expression \( y = x + 2 \) represents a linear function. Let's break down what makes it a linear function:

Characteristics of a Linear Function

1. Form: A linear function can generally be expressed in the form \( y = mx + b \), where:

   • \( m \) is the slope of the line,
   • \( b \) is the y-intercept, which is where the line crosses the y-axis.

2. In your function:

   • The equation \( y = x + 2 \) can be seen as:
     • Slope (\( m \)): 1 (since there is an implicit \( 1 \) in front of \( x \)),
     • Y-intercept (\( b \)): 2 (the value of \( y \) when \( x = 0 \)).

3. Graph:

   • The graph of this linear function is a straight line with a slope of 1 that crosses the y-axis at the point (0, 2).

Summary

So, indeed, \( y = x + 2 \) is a linear function. If you have further questions or need additional explanations, feel free to ask!

Answer 8

9. The graph of a function, f(x), is shown.

Find the value of f(-3).

(1 point)
f(−3)=

Answer 9

Unfortunately, I can't see the graph you mentioned, so I am unable to directly find the value of \( f(-3) \). However, I can guide you on how to find \( f(-3) \) using a graph:

1. Find -3 on the x-axis: Look for the point where \( x = -3 \) on the horizontal axis (x-axis).

2. Determine the corresponding y-value: Once you find \( x = -3 \), look vertically upward (or downward) to find where the graph intersects or meets the vertical line at \( x = -3 \). The y-coordinate at this intersection point will give you the value of \( f(-3) \).

If you describe the graph or provide additional details, I'd be happy to assist you further!

Answer 10

13. A soccer player kicks a ball with an initial velocity of 38 feet every second. The function f(x)=−16x2+38x
can be used to model the time the ball is in the air.
What is the meaning of the origin for this situation?(1 point)
Responses

After 38 seconds, the ball is at a height of 16 feet.
After 38 seconds, the ball is at a height of 16 feet.

After 16 seconds, the ball is at a height of 38 feet.
After 16 seconds, the ball is at a height of 38 feet.

At 0 seconds, the ball is at a height of 38 feet.
At 0 seconds, the ball is at a height of 38 feet.

At 0 seconds, the ball is at a height of 0 feet.

Answer 11

To analyze the function \( f(x) = -16x^2 + 38x \) which models the height of the soccer ball over time, let's interpret the origin (\( x = 0 \)) in this context:

1. Function Analysis:

   • The variable \( x \) represents time in seconds after the soccer ball is kicked.
   • The output of the function, \( f(x) \), represents the height of the ball in feet at any time \( x \).

2. Meaning of the Origin:

   • At \( x = 0 \) (which represents the moment the ball is kicked), substituting \( x = 0 \) into the function gives us: \[ f(0) = -16(0)^2 + 38(0) = 0. \]
   • This means that at time \( t = 0 \) seconds, the height of the ball is \( 0 \) feet, indicating it starts at ground level when kicked.

Conclusion

Therefore, the correct interpretation of the origin (\( x = 0 \)) in this situation is:

At 0 seconds, the ball is at a height of 0 feet.

So the correct response is:

At 0 seconds, the ball is at a height of 0 feet.