Question

what is the rate of change of the function passing through the points (3, 15) and (-6, 3)?
○ \\(\frac{3}{4}\\)
○ \\(-\frac{4}{3}\\)
○ \\(\frac{4}{3}\\)
○ -4

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Let \((x_1, y_1)=(3, 15)\) and \((x_2, y_2)=(-6, 3)\).

Step2: Substitute values into formula

Substitute \(x_1 = 3\), \(y_1 = 15\), \(x_2=-6\), \(y_2 = 3\) into the formula:
\(m=\frac{3 - 15}{-6 - 3}\)

Step3: Simplify numerator and denominator

Calculate numerator: \(3 - 15=-12\)
Calculate denominator: \(-6 - 3=-9\)
So \(m=\frac{-12}{-9}\)

Step4: Reduce the fraction

Simplify \(\frac{-12}{-9}\) by dividing numerator and denominator by their greatest common divisor (3):
\(\frac{-12\div3}{-9\div3}=\frac{4}{3}\)? Wait, no, wait. Wait, let's re - check. Wait, \(y_2 - y_1=3 - 15=-12\), \(x_2 - x_1=-6 - 3=-9\). Then \(\frac{-12}{-9}=\frac{12}{9}=\frac{4}{3}\)? But that's not matching the selected option. Wait, maybe I mixed up the points. Let's swap the points. Let \((x_1,y_1)=(-6,3)\) and \((x_2,y_2)=(3,15)\). Then \(m=\frac{15 - 3}{3-(-6)}=\frac{12}{9}=\frac{4}{3}\)? No, the selected option is \(-\frac{4}{3}\). Wait, I must have made a mistake. Wait, the formula is \(\frac{y_2 - y_1}{x_2 - x_1}\). Let's take the first point as \((3,15)\) and the second as \((-6,3)\). So \(y_2 - y_1=3 - 15=-12\), \(x_2 - x_1=-6 - 3=-9\). Then \(\frac{-12}{-9}=\frac{4}{3}\). But the selected option is \(-\frac{4}{3}\). Wait, maybe the user made a mistake in the selected option? Wait, no, let's recalculate. Wait, \(x_2 - x_1=-6 - 3=-9\), \(y_2 - y_1=3 - 15=-12\). So \(\frac{-12}{-9}=\frac{4}{3}\). But the options have \(\frac{4}{3}\) as an option (the third option). Wait, the blue - circled option is \(-\frac{4}{3}\), but according to the calculation, it's \(\frac{4}{3}\). Wait, maybe I messed up the order of subtraction. Let's use the formula \(\frac{y_1 - y_2}{x_1 - x_2}\). Then \(y_1 - y_2=15 - 3 = 12\), \(x_1 - x_2=3-(-6)=9\), \(\frac{12}{9}=\frac{4}{3}\). Alternatively, if we do \(\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 15}{-6 - 3}=\frac{-12}{-9}=\frac{4}{3}\). So the correct rate of change is \(\frac{4}{3}\), which is the third option (the unselected one with \(\frac{4}{3}\)). But the user's selected option is \(-\frac{4}{3}\), which is incorrect. But according to the calculation, the rate of change is \(\frac{4}{3}\). Wait, maybe there is a miscalculation. Wait, \(3-15=-12\), \(-6 - 3=-9\), \(\frac{-12}{-9}=\frac{4}{3}\). Yes, that's correct. So the correct answer should be \(\frac{4}{3}\), which is the option " \(\boldsymbol{\frac{4}{3}}\)".

Answer:

\(\frac{4}{3}\) (The option with \(\frac{4}{3}\))