Question

1. the table shows the relationship between the number of photos x you take and the amount of memory y in megabytes (mb) left on your cameras memory chip. is the relationship a linear function? describe the relationship

Explanation:

Step1: Calculate the rate of change (slope)

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \((x_1,y_1)=(0,512)\) and \((x_2,y_1)=(1,509)\), \(m_1=\frac{509 - 512}{1-0}=\frac{- 3}{1}=-3\).
For \((x_1,y_1)=(1,509)\) and \((x_2,y_1)=(2,506)\), \(m_2=\frac{506 - 509}{2 - 1}=\frac{-3}{1}=-3\).
For \((x_1,y_1)=(2,506)\) and \((x_2,y_1)=(3,503)\), \(m_3=\frac{503 - 506}{3 - 2}=\frac{-3}{1}=-3\).

Step2: Check if it is a linear function

Since the rate of change (slope) \(m=-3\) is constant for all pairs of points. A linear function has the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. Here, when \(x = 0\), \(y=512\), so \(b = 512\). The equation of the function is \(y=-3x + 512\)

Answer:

Yes, the relationship is a linear function. The equation of the linear function is \(y=-3x + 512\), which means that for each additional photo taken (\(x\) increases by 1), the memory left (\(y\)) decreases by 3 megabytes. The initial memory (when \(x = 0\)) is 512 megabytes.