Question

the table shows the coordinates of some points on the quadratic function, g (x). a student uses the table to calculate the average rate of change of the function over the interval 0,6. the student’s starting expression to set up the calculation is also shown. explain whether the student’s starting expression is correct. use the drop - down menus to complete the statements. function g(x) | x | -2 | 0 | 3 | 6 | 10 | | g(x) | -3 | -7 | -8 | -3 | 13 | student’s starting expression: \\(\frac{-7 + (-3)}{2}\\) click the arrows to choose an answer from each menu. the expression to calculate the average rate of change over the interval 0,6 is the choose... . the average rate of change over the specified interval is choose... . so, the student’s starting expression is choose... .

Explanation:

Step1: Recall average rate of change formula

The formula for the average rate of change of a function \( g(x) \) over the interval \([a, b]\) is \(\frac{g(b) - g(a)}{b - a}\).

Step2: Identify \( a \), \( b \), \( g(a) \), \( g(b) \)

For the interval \([0, 6]\), \( a = 0 \), \( b = 6 \). From the table, \( g(0)= - 7 \) and \( g(6)= - 3 \).

Step3: Calculate the correct expression

Substitute into the formula: \(\frac{g(6)-g(0)}{6 - 0}=\frac{-3-(-7)}{6-0}=\frac{-3 + 7}{6}=\frac{4}{6}=\frac{2}{3}\).

Step4: Analyze the student's expression

The student's expression is \(\frac{-7+(-3)}{2}=\frac{-10}{2}=- 5\), which is incorrect. The student used the average of \( g(0) \) and \( g(6) \) instead of the average rate of change formula.

1. The formula for the average rate of change over \([a,b]\) is \(\frac{g(b)-g(a)}{b - a}\). For \([0,6]\), \(a = 0\), \(b = 6\), \(g(0)=-7\), \(g(6)=-3\).
2. Correct expression: \(\frac{g(6)-g(0)}{6 - 0}=\frac{-3-(-7)}{6}=\frac{4}{6}=\frac{2}{3}\).
3. Student's expression: \(\frac{-7+(-3)}{2}\) is wrong as it's the average of \(g(0)\) and \(g(6)\), not the rate of change.

Answer:

• The expression to calculate the average rate of change over the interval \([0,6]\) is \(\boldsymbol{\frac{g(6)-g(0)}{6 - 0}}\) (or \(\boldsymbol{\frac{-3-(-7)}{6}}\)).
• The average rate of change over the specified interval is \(\boldsymbol{\frac{2}{3}}\).
• So, the student's starting expression is \(\boldsymbol{\text{incorrect}}\).