Question

Keylie is a freelance web designer. She charges her clients a $20 consulting fee plus $45 per hour worked. If they need images, Keylie charges an additional $15 fee for image research. Create a graph that represents how much money Keylie will make if she has to work 4, 6, 8, 10, and 12 hours for a client and has to find images. Make sure to label your axis and use an appropriate scale of 100 for your y-axis, and 5 for your x-axis.

Answer 1

To represent Keylie's earnings based on the hours worked, we can create an equation that calculates her total charge for a client. The total charge can be computed using the following formula:

Total Charge = Consulting Fee + (Hourly Rate × Hours Worked) + Image Research Fee

Given:

• Consulting Fee = $20
• Hourly Rate = $45
• Image Research Fee = $15

The formula becomes:

Total Charge = 20 + (45 × Hours Worked) + 15

This simplifies to:

Total Charge = 35 + 45 × Hours Worked

Now, we can calculate her total earnings for the specified hours: 4, 6, 8, 10, and 12.

1. For 4 hours: Total Charge = 35 + 45 × 4 = 35 + 180 = $215

2. For 6 hours: Total Charge = 35 + 45 × 6 = 35 + 270 = $305

3. For 8 hours: Total Charge = 35 + 45 × 8 = 35 + 360 = $395

4. For 10 hours: Total Charge = 35 + 45 × 10 = 35 + 450 = $485

5. For 12 hours: Total Charge = 35 + 45 × 12 = 35 + 540 = $575

Summary of Earnings

• 4 hours: $215
• 6 hours: $305
• 8 hours: $395
• 10 hours: $485
• 12 hours: $575

Graph Description

Now that we have the data, we can create a graph:

1. X-axis: Represents the hours worked (0, 4, 6, 8, 10, 12) with a scale of 5.
2. Y-axis: Represents the total charge (total earnings) with a scale of 100.

Points to Plot

• (4, 215)
• (6, 305)
• (8, 395)
• (10, 485)
• (12, 575)

Sketching the Graph

1. Draw a vertical line for the Y-axis and label it "Total Charge ($)".
2. Draw a horizontal line for the X-axis and label it "Hours Worked".
3. Mark the Y-axis at intervals of 100 up to 600 (0, 100, 200, 300, 400, 500, 600).
4. Mark the X-axis at intervals of 5 up to 15 (0, 5, 10, 15).
5. Plot the points according to the calculated earnings.
6. Connect the points with a line to represent the relationship between hours worked and total charge.

Conclusion

This graph shows the linear relationship between the hours worked and the total charge, indicating that as the number of hours increases, the total charge increases correspondingly, reflecting Keylie's pricing structure.

Answer 2

Create a scenario given the information provided. Use 3-5 sentences to describe your scenario in which you would use the given information (function or graph) and re-write or write the function and explain the meaning of each variable.

Quadratic functions can be used to describe the rate at which an object will fall and predict when it will land or how high it may go. Create a quadratic function using the function to help predict when an object will return to the ground from being kicked, dropped, or thrown.

Answer 3

Scenario

Imagine you are coaching a youth soccer team, and you want to explain to your players how to calculate the height of a soccer ball that a player has kicked into the air. The ball is kicked from a height of 2 feet with an initial velocity of 20 feet per second. As the ball rises, it is affected by gravity, causing it to reach a maximum height before eventually falling back down. By understanding the quadratic function that represents the ball's height over time, players can better grasp how long it will stay in the air and when to anticipate its landing.

Quadratic Function

The quadratic function to describe the height of the soccer ball over time can be expressed as:

\[ h(t) = -16t^2 + 20t + 2 \]

Explanation of Variables

• \( h(t) \): This is the height of the soccer ball (in feet) at time \( t \) seconds.
• \( t \): This represents the time (in seconds) since the ball was kicked.
• -16: This coefficient represents the effect of gravity on the ball's height. It shows that the height is decreasing over time due to gravitational pull (the approximate acceleration due to gravity in feet per second squared divided by 2).
• 20: This is the initial velocity of the ball in feet per second. It indicates the speed at which the ball was kicked upwards.
• 2: This is the initial height from which the ball was kicked (in feet). It shows that the ball starts 2 feet above the ground.

Conclusion

Using this quadratic function, we can predict various aspects of the ball's motion, such as its maximum height, the time it takes to reach that height, and when it will return to the ground. This understanding is vital for players as they learn to anticipate the ball’s trajectory during gameplay.