Question
Jasmin modeled the relationship between the altitude of a tree
\[A\] (in meters above sea level) and the tree's growth rate
\[R\] (in millimeters per year) as
\[R=-0.03A^2+0.42A+1.13\].
She wanted to graph the relationship for altitudes between
\[0\] and
\[9000\] meters. Here is her work:
A concave down parabola on an x y coordinate plane. The y axis scales by four tenths from zero to two and eight tenths. The x axis scales by one thousand from zero to nine thousand. The y axis is labeled Growth rate in millimeters per year. The x axis is labeled altitude in meters above sea level. The parabola starts at (zero, one and thirteen hundredths). It increases up to a maximum around (seven thousand, two and six tenths) before decreasing.
\[\small{2000}\]
\[\small{4000}\]
\[\small{6000}\]
\[\small{8000}\]
\[\small{0.8}\]
\[\small{1.6}\]
\[\small{2.4}\]
What mistakes did Jasmin make when drawing the graph?
Choose all answers that apply:
Choose all answers that apply:
(Choice A) Inappropriate scale for
\[x\]-axis
A
Inappropriate scale for
\[x\]-axis
(Choice B) Inappropriate scale for
\[y\]-axis
B
Inappropriate scale for
\[y\]-axis
(Choice C) Wrong/missing label for
\[x\]-axis
C
Wrong/missing label for
\[x\]-axis
(Choice D) Wrong/missing label for
\[y\]-axis
D
Wrong/missing label for
\[y\]-axis
(Choice E) Jasmin didn't make any mistake
E
Jasmin didn't make any mistake
\[A\] (in meters above sea level) and the tree's growth rate
\[R\] (in millimeters per year) as
\[R=-0.03A^2+0.42A+1.13\].
She wanted to graph the relationship for altitudes between
\[0\] and
\[9000\] meters. Here is her work:
A concave down parabola on an x y coordinate plane. The y axis scales by four tenths from zero to two and eight tenths. The x axis scales by one thousand from zero to nine thousand. The y axis is labeled Growth rate in millimeters per year. The x axis is labeled altitude in meters above sea level. The parabola starts at (zero, one and thirteen hundredths). It increases up to a maximum around (seven thousand, two and six tenths) before decreasing.
\[\small{2000}\]
\[\small{4000}\]
\[\small{6000}\]
\[\small{8000}\]
\[\small{0.8}\]
\[\small{1.6}\]
\[\small{2.4}\]
What mistakes did Jasmin make when drawing the graph?
Choose all answers that apply:
Choose all answers that apply:
(Choice A) Inappropriate scale for
\[x\]-axis
A
Inappropriate scale for
\[x\]-axis
(Choice B) Inappropriate scale for
\[y\]-axis
B
Inappropriate scale for
\[y\]-axis
(Choice C) Wrong/missing label for
\[x\]-axis
C
Wrong/missing label for
\[x\]-axis
(Choice D) Wrong/missing label for
\[y\]-axis
D
Wrong/missing label for
\[y\]-axis
(Choice E) Jasmin didn't make any mistake
E
Jasmin didn't make any mistake
Answers
GPT-4o mini
To determine the mistakes made by Jasmin in the graph of the relationship between altitude \( A \) and growth rate \( R \), we can analyze the given information about the quadratic function and how it applies to the graph she created.
1. **Inappropriate scale for \( x \)-axis (Choice A)**: The \( x \)-axis is labeled as representing altitude in meters above sea level, and it scales from 0 to 9000 meters. This scale appears appropriate since it covers the intended range. Thus, **this choice is not a mistake**.
2. **Inappropriate scale for \( y \)-axis (Choice B)**: The quadratic function \( R = -0.03A^2 + 0.42A + 1.13 \) suggests that the growth rate \( R \) can have a maximum and then decrease, potentially exceeding the maximum values noted on the \( y \)-axis if not scaled properly. You need to calculate the vertex (maximum) to assess whether the \( y \)-axis appropriately covers all values of \( R \).
First, calculate the vertex \( A \) value:
\[
A = -\frac{b}{2a} = -\frac{0.42}{2 \times -0.03} = 7.0 \text{ (this is 7000 meters)}
\]
Now substitute \( A = 7000 \) in the function to find the maximum growth rate \( R \):
\[
R = -0.03(7000)^2 + 0.42(7000) + 1.13 = -0.03(49000000) + 2940 + 1.13
\]
\[
R = -1470000 + 2940 + 1.13 \approx 1.14 \text{ millimeters per year}
\]
The highest value of \( R \) reached in the general range provided seems to be below 2.8 (since the graph goes up to 2.8 in the y-axis). Hence, **the scale for the \( y \)-axis is appropriate though slightly compressed to visualize the maximum**.
3. **Wrong/missing label for \( x \)-axis (Choice C)**: The \( x \)-axis is clearly labeled as "altitude in meters above sea level," which is appropriate. Therefore, **this choice is not a mistake**.
4. **Wrong/missing label for \( y \)-axis (Choice D)**: The \( y \)-axis is labeled "Growth rate in millimeters per year," which accurately describes what is being represented. Therefore, **this choice is also not a mistake**.
5. **Jasmin didn't make any mistake (Choice E)**: Given that the explanations for the choices above clarify that there are no significant mistakes related to the axes, labeling, or scale, this option would be considered the most accurate conclusion.
Based on the analysis of the information given and the relationships stated, **the correct choice is E (Jasmin didn't make any mistake)**.
1. **Inappropriate scale for \( x \)-axis (Choice A)**: The \( x \)-axis is labeled as representing altitude in meters above sea level, and it scales from 0 to 9000 meters. This scale appears appropriate since it covers the intended range. Thus, **this choice is not a mistake**.
2. **Inappropriate scale for \( y \)-axis (Choice B)**: The quadratic function \( R = -0.03A^2 + 0.42A + 1.13 \) suggests that the growth rate \( R \) can have a maximum and then decrease, potentially exceeding the maximum values noted on the \( y \)-axis if not scaled properly. You need to calculate the vertex (maximum) to assess whether the \( y \)-axis appropriately covers all values of \( R \).
First, calculate the vertex \( A \) value:
\[
A = -\frac{b}{2a} = -\frac{0.42}{2 \times -0.03} = 7.0 \text{ (this is 7000 meters)}
\]
Now substitute \( A = 7000 \) in the function to find the maximum growth rate \( R \):
\[
R = -0.03(7000)^2 + 0.42(7000) + 1.13 = -0.03(49000000) + 2940 + 1.13
\]
\[
R = -1470000 + 2940 + 1.13 \approx 1.14 \text{ millimeters per year}
\]
The highest value of \( R \) reached in the general range provided seems to be below 2.8 (since the graph goes up to 2.8 in the y-axis). Hence, **the scale for the \( y \)-axis is appropriate though slightly compressed to visualize the maximum**.
3. **Wrong/missing label for \( x \)-axis (Choice C)**: The \( x \)-axis is clearly labeled as "altitude in meters above sea level," which is appropriate. Therefore, **this choice is not a mistake**.
4. **Wrong/missing label for \( y \)-axis (Choice D)**: The \( y \)-axis is labeled "Growth rate in millimeters per year," which accurately describes what is being represented. Therefore, **this choice is also not a mistake**.
5. **Jasmin didn't make any mistake (Choice E)**: Given that the explanations for the choices above clarify that there are no significant mistakes related to the axes, labeling, or scale, this option would be considered the most accurate conclusion.
Based on the analysis of the information given and the relationships stated, **the correct choice is E (Jasmin didn't make any mistake)**.