Question

which table shows a linear function?
x: 0, 1, 2, 3
y: 5, 8, 11, 14
which equation represents the function?
y = 5x + 3, y = 3x + 5, y = x + 5, y = x + 3

Explanation:

First Sub - Question (Which table shows a linear function?)

Step1: Recall linear function property

A linear function has a constant rate of change (slope). For a table of values, the difference in \(y\) - values ( \(\Delta y\)) should be constant for equal differences in \(x\) - values ( \(\Delta x\)). Here, \(\Delta x=1 - 0 = 1\), \(2 - 1 = 1\), \(3 - 2 = 1\) (constant \(\Delta x = 1\)).

Step2: Calculate \(\Delta y\)

For the given table: When \(x\) changes from \(0\) to \(1\), \(y\) changes from \(5\) to \(8\), \(\Delta y=8 - 5 = 3\). When \(x\) changes from \(1\) to \(2\), \(y\) changes from \(8\) to \(11\), \(\Delta y = 11-8 = 3\). When \(x\) changes from \(2\) to \(3\), \(y\) changes from \(11\) to \(14\), \(\Delta y=14 - 11 = 3\). Since \(\Delta y\) is constant (\(3\)) for \(\Delta x = 1\), the table with \(x:0,1,2,3\) and \(y:5,8,11,14\) shows a linear function.

Second Sub - Question (Which equation represents the function?)

Step1: Recall slope - intercept form

The slope - intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept (value of \(y\) when \(x = 0\)).

Step2: Find \(b\)

From the table, when \(x = 0\), \(y = 5\). So, \(b = 5\).

Step3: Find \(m\) (slope)

We know that \(m=\frac{\Delta y}{\Delta x}\). We calculated \(\Delta y = 3\) and \(\Delta x = 1\), so \(m = 3\).

Step4: Write the equation

Substitute \(m = 3\) and \(b = 5\) into \(y=mx + b\). We get \(y=3x + 5\).

Answer:

• The table with \(x\) values \(0,1,2,3\) and \(y\) values \(5,8,11,14\) shows a linear function.
• The equation that represents the function is \(y = 3x+5\)