Math 143 Week 5 Desmos Activity Worksheet
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School
Boise State University *
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Course
143
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
3
Uploaded by AgentCrownDove34
M143 Week 5 Worksheet Math 143 Week 5 Activity Worksheet Quadratic Regression Directions: These activity questions are written to follow the Desmos activity for the week. Once you have completed the weekly activity you will be able to answer these questions. You can complete the questions on any document and then hand in the document on Canvas. 1.
Given the quadratic equation y
= 3
x
2
–
6
x
+ 1 , explain how to find the x-intercepts both graphically and algebraically. Find the x-intercepts. To find the x-intercepts graphically, we plot the quadratic equation on a graph and identify the points where the graph crosses the x-axis. y=3x 2 −6x+1 are approximately: x
≈ 0.219 and x
≈ 2.781 x≈0.219andx≈2.781
2.
Given the quadratic function y
= 3
x
2
–
6
x
+ 1, explain how you would find the vertex both algebraically and graphically. Find the vertex. From the graph, you can see that the vertex is at the point ( 1 , − 2 ) (1,−2), which aligns with our algebraic calculation. 3.
What is the purpose of the discriminant? If the discriminant value is 16, what do you know about the equation and the graph? The discriminant tells us about the roots of a quadratic equation: If it's positive (like 16), the equation has two real roots. The graph of the equation is a U-shape, opening upwards, and it crosses the x-axis at two points. 4.
A ball is thrown straight up in the air. The table shows the height h
, in feet of the ball after t
seconds. time (
t
) in seconds 0 1 2 3 4 5 6 height (
h
) in feet 5 94 146 166 154 109 34
M143 Week 5 Worksheet a)
Plot the data on Desmos. Sketch your plot below. Make sure to label your vertical and horizontal axis. b)
What is you window? Explain. The window settings for the plot would typically depend on the range of values in the data set. Since the time t values start from 0 and go up to 6 seconds, and the height ℎ h values range from 5 feet to 166 feet c)
What type of regression would you use for the data? Justify your choice. A quadratic regression is a good fit for the data because when a ball is thrown up and falls back down, its motion follows a curved path like a parabola. Quadratic regression finds the best-fit parabola that describes this motion accurately, helping us understand how the ball's height changes over time. d)
On Desmos, find the quadratic regression that best fits the data. Write your equation below. h(t)=at
2
+bt+c e)
Does your graph have a maximum or minimum value? Explain. Yes, the graph of the ball's height has either a highest point (maximum) or lowest point (minimum) because the ball goes up and then comes down due to gravity.
M143 Week 5 Worksheet f)
Find the maximum or minimum value algebraically. To find this highest or lowest point algebraically, we find the vertex of the quadratic function representing the ball's height. This is done by using a formula and the coefficients obtained from regression analysis, which helps us find the exact time and height when the ball reaches its highest or lowest point. g)
Find the maximum or minimum using Desmos. t=3, h=50 h)
Explain what the maximum/minimum value means in the context of the problem. The maximum value represents the highest point the ball reaches before falling back down due to gravity. It's like the peak of its flight. The minimum value, if applicable, would be the lowest point the ball reaches before hitting the ground again.
i)
Find the x
-intercepts of the graph using Desmos. t~3+
√
10
j)
Explain what the x
-intercepts mean in the context of the problem. The x-intercepts in this problem represent the times when the ball hits the ground
—
once when it's initially thrown and again when it falls back down after reaching its maximum height. k)
Explain how you would find the height of the ball at 8 seconds both algebraically and graphically. Find the height of the ball at 8 seconds algebraically. To find the height of the ball at 8 seconds: Algebraically: We use the quadratic equation representing the ball's height and substitute t
= 8 to find the height. Graphically: We locate the point at t=8 on the graph of the quadratic function and read the corresponding height value.
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