Question

3 the function represented by the table:
input | output
10 | 12
5 | 9
-5 | 3

Explanation:

Step 1: Identify two points

From the table, we have two points: when input \( x_1 = 10 \), output \( y_1 = 12 \); when input \( x_2 = 5 \), output \( y_2 = 9 \); or when \( x_3=- 5\), output \( y_3 = 3\). Let's use \((10,12)\) and \((5,9)\) to find the slope \( m\) of the linear function \( y=mx + b\). The formula for slope is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
\(m=\frac{9 - 12}{5 - 10}=\frac{-3}{-5}=\frac{3}{5}\)? Wait, no, let's check with \((10,12)\) and \((- 5,3)\). \(m=\frac{3 - 12}{-5 - 10}=\frac{-9}{-15}=\frac{3}{5}\)? Wait, maybe I made a mistake. Wait, let's take \((10,12)\) and \((5,9)\): the difference in \(y\) is \(9 - 12=-3\), difference in \(x\) is \(5 - 10=-5\), so \(m=\frac{-3}{-5}=\frac{3}{5}\). Then use point - slope form \(y - y_1=m(x - x_1)\). Using \((10,12)\): \(y-12=\frac{3}{5}(x - 10)\). \(y-12=\frac{3}{5}x-6\), so \(y=\frac{3}{5}x + 6\). Wait, but let's check with \(x = 5\): \(y=\frac{3}{5}\times5+6=3 + 6 = 9\), which matches. \(x=-5\): \(y=\frac{3}{5}\times(-5)+6=-3 + 6 = 3\), which also matches. So the function is \(y=\frac{3}{5}x + 6\). But maybe the problem is to find the equation of the linear function. Let's re - examine the table. Wait, maybe the input and output are mixed? Wait, the table has input: 10, 5, - 5 and output:12, 9, 3. So the pairs are \((10,12)\), \((5,9)\), \((-5,3)\). Let's find the slope between \((10,12)\) and \((5,9)\): \(m=\frac{9 - 12}{5 - 10}=\frac{-3}{-5}=\frac{3}{5}\). Then the equation is \(y - 12=\frac{3}{5}(x - 10)\), \(y=\frac{3}{5}x-6 + 12\), \(y=\frac{3}{5}x + 6\). Let's check \(x = 5\): \(y=\frac{3}{5}\times5+6=3 + 6 = 9\) (correct). \(x=-5\): \(y=\frac{3}{5}\times(-5)+6=-3 + 6 = 3\) (correct).

Step 2: If the problem is to find the slope

The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Taking \((x_1,y_1)=(10,12)\) and \((x_2,y_2)=(-5,3)\), we have \(m=\frac{3 - 12}{-5 - 10}=\frac{-9}{-15}=\frac{3}{5}\). If we take \((x_1,y_1)=(5,9)\) and \((x_2,y_2)=(-5,3)\), \(m=\frac{3 - 9}{-5 - 5}=\frac{-6}{-10}=\frac{3}{5}\). So the slope of the linear function is \(\frac{3}{5}\).

Answer:

If the problem is to find the equation of the linear function, the equation is \(y=\frac{3}{5}x + 6\). If it is to find the slope, the slope is \(\frac{3}{5}\). (Assuming the problem is about finding the linear function equation or slope from the table, the final answer depends on the specific question, but based on the table, the linear function is \(y=\frac{3}{5}x + 6\))