Question

find the equation of the linear function represented by the table below in slope intercept form.

Explanation:

Step1: Identify two points

Let's take two points from the table, say \((x_1, y_1)=(- 8,3)\) and \((x_2, y_2)=(3,7)\).

Step2: Calculate the slope \(m\)

The formula for slope is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute the values: \(m=\frac{7 - 3}{3-(-8)}=\frac{4}{11}\)? Wait, no, wait, maybe I misread the table. Wait, let's check the table again. Wait, maybe the table is: let's assume the first column is \(x\) and the second is \(y\). Let's take another pair. Wait, maybe the points are \((-8,3)\), \((3,7)\), \((9,17)\), \((12,27)\)? Wait, no, when \(x = 3\), \(y=7\); \(x = 9\), \(y = 17\). Let's recalculate slope between \((3,7)\) and \((9,17)\). \(m=\frac{17 - 7}{9 - 3}=\frac{10}{6}=\frac{5}{3}\)? No, that doesn't seem right. Wait, maybe the table is: let's check the differences. From \(x=-8\) to \(x = 3\), the change in \(x\) is \(3-(-8)=11\), change in \(y\) is \(7 - 3=4\). From \(x = 3\) to \(x=9\), change in \(x\) is \(6\), change in \(y\) is \(17 - 7 = 10\). From \(x=9\) to \(x = 12\), change in \(x\) is \(3\), change in \(y\) is \(27 - 17=10\)? Wait, no, maybe the table is written with some typos. Wait, maybe the correct points are \((-8,3)\), \((3,7)\), \((9,17)\), \((12,27)\) is wrong. Wait, let's try again. Let's take \((x_1,y_1)=(-8,3)\) and \((x_2,y_2)=(3,7)\). Then \(m=\frac{7 - 3}{3+8}=\frac{4}{11}\). Then take \((3,7)\) and \((9,17)\): \(m=\frac{17 - 7}{9 - 3}=\frac{10}{6}=\frac{5}{3}\). These are different, so maybe the table is miswritten. Wait, maybe the \(y\)-values are 3, 7, 13, 17? No, the last one is 27. Wait, maybe the correct slope is 2? Let's check: if \(x=-8\), \(y = 3\); if \(m = 2\), then \(y=2x + b\). Plug \(x=-8\), \(y=3\): \(3=2*(-8)+b\) => \(3=-16 + b\) => \(b = 19\). Then check \(x = 3\): \(y=2*3+19=6 + 19=25\), but the table says 7. So that's wrong. Wait, maybe the table is:

| \(x\) | \(y\) |
|------|------|
| -8 | 3 |
| 3 | 7 |
| 9 | 17 |
| 12 | 27 |

Wait, let's calculate the slope between \((-8,3)\) and \((3,7)\): \(m=\frac{7 - 3}{3 - (-8)}=\frac{4}{11}\approx0.36\). Between \((3,7)\) and \((9,17)\): \(m=\frac{17 - 7}{9 - 3}=\frac{10}{6}=\frac{5}{3}\approx1.67\). Between \((9,17)\) and \((12,27)\): \(m=\frac{27 - 17}{12 - 9}=\frac{10}{3}\approx3.33\). These are not consistent, so maybe the table has errors. Alternatively, maybe the \(x\)-values are -8, -3, 2, 7? No, the user's table is as given. Wait, maybe the correct approach is to assume it's a linear function, so the slope should be constant. Let's check the differences in \(x\) and \(y\). Let's list the \(x\) values: -8, 3, 9, 12. The differences between \(x\) values: 3 - (-8)=11, 9 - 3 = 6, 12 - 9=3. The \(y\) values: 3,7,17,27. Differences: 7 - 3=4, 17 - 7=10, 27 - 17=10. Not constant. So maybe the table is miswritten. Wait, maybe the \(y\)-value for \(x = 3\) is 13? Then 13 - 3=10, 17 - 13=4, no. Alternatively, maybe the \(x\)-value for the last one is 14? No. Wait, maybe the original problem has a different table. Wait, perhaps the correct table is:

| \(x\) | \(y\) |
|------|------|
| -8 | 3 |
| 3 | 13 |
| 9 | 23 |
| 12 | 29 |

Then slope between (-8,3) and (3,13) is \(\frac{13 - 3}{3 + 8}=\frac{10}{11}\). No. Alternatively, maybe the user made a typo. Wait, let's try to proceed with the given table as is, assuming that maybe it's a linear function with slope \(m = \frac{4}{11}\) first. Then use point-slope form \(y - y_1=m(x - x_1)\). Using \((-8,3)\): \(y - 3=\frac{4}{11}(x + 8)\). Then \(y=\frac{4}{11}x+\frac{32}{11}+3=\frac{4}{11}x+\frac{32 + 33}{11}=\frac{4}{11}x+\frac{65}{11}\). But when \(x = 3\), \(y=\frac{12}{11}+\frac{65}{11}=\frac{77}{11}=7\), which matches. When \(x = 9\), \(y=\frac{36}{11}+\frac{65}{11}=\frac{101}{11}\approx9.18\), but the table says 17. So that's wrong. So maybe the slope is 2. Wait, if \(m = 2\), then \(y=2x + b\). For \(x=-8\), \(y=3\): \(3=2*(-8)+b\) => \(b = 19\). Then \(x = 3\): \(y=6 + 19=25 eq7\). No. Wait, maybe the slope is 1. Then \(y=x + b\). For \(x=-8\), \(y=3\): \(3=-8 + b\) => \(b = 11\). Then \(x = 3\): \(y=3 + 11=14 eq7\). No. Wait, maybe the slope is \(\frac{2}{3}\). Then \(y=\frac{2}{3}x + b\). For \(x=-8\), \(y=3\): \(3=\frac{2}{3}*(-8)+b\) => \(3=-\frac{16}{3}+b\) => \(b=3+\frac{16}{3}=\frac{25}{3}\). Then \(x = 3\): \(y=2+\frac{25}{3}=\frac{31}{3}\approx10.33 eq7\). No. Wait, maybe the table is:

| \(x\) | \(y\) |
|------|------|
| -8 | 3 |
| 3 | 7 |
| 9 | 17 |
| 14 | 27 |

Then slope between (-8,3) and (3,7): \(\frac{4}{11}\), between (3,7) and (9,17): \(\frac{10}{6}=\frac{5}{3}\), between (9,17) and (14,27): \(\frac{10}{5}=2\). Still inconsistent. Wait, maybe the original problem has a different table. Alternatively, maybe the \(y\)-values are 3, 7, 13, 17? No, the last one is 27. Wait, maybe the correct table is with \(x=-8,y=3\); \(x=3,y=13\); \(x=9,y=23\); \(x=14,y=33\). Then slope is 2. Then \(y=2x + b\), \(3=2*(-8)+b\) => \(b=19\), \(y=2x + 19\). Then \(x=3\), \(y=6 + 19=25 eq13\). No. I think there might be a typo in the table. But assuming the table is correct as given, and using the first two points \((-8,3)\) and \((3,7)\) to find the slope, then the equation is \(y=\frac{4}{11}x+\frac{65}{11}\). But when we check \(x=9\), it doesn't match. Alternatively, maybe the table is:

| \(x\) | \(y\) |
|------|------|
| -8 | 3 |
| 3 | 7 |
| 9 | 19 |
| 12 | 25 |

Then slope between (-8,3) and (3,7) is \(\frac{4}{11}\), between (3,7) and (9,19) is \(\frac{12}{6}=2\). Still inconsistent. I think the most probable is that there is a typo, but if we proceed with the first two points, the slope is \(\frac{4}{11}\) and the equation is \(y=\frac{4}{11}x+\frac{65}{11}\). But maybe the intended table is with \(y\)-values 3, 13, 23, 33, then slope 2, equation \(y=2x + 19\). But given the table as is, let's use the first two points.

Step3: Find the y-intercept \(b\)

Using the slope \(m=\frac{4}{11}\) and the point \((-8,3)\) in \(y=mx + b\).

\(3=\frac{4}{11}*(-8)+b\)

\(3=-\frac{32}{11}+b\)

\(b=3+\frac{32}{11}=\frac{33 + 32}{11}=\frac{65}{11}\)

Step4: Write the equation

The slope-intercept form is \(y=mx + b\), so \(y=\frac{4}{11}x+\frac{65}{11}\). But when we check \(x = 9\), \(y=\frac{36}{11}+\frac{65}{11}=\frac{101}{11}\approx9.18\), which is not 17. So there must be a typo. Alternatively, maybe the correct slope is 2, and the \(y\)-values are 3, 13, 23, 33. Then the equation is \(y=2x + 19\). But given the table as provided, with the points \((-8,3)\), \((3,7)\), \((9,17)\), \((12,27)\), the slope between \((3,7)\) and \((9,17)\) is \(\frac{10}{6}=\frac{5}{3}\), between \((9,17)\) and \((12,27)\) is \(\frac{10}{3}\). These are not linear. So I think there is a mistake in the table. But if we assume that the table is correct and it's a linear function, maybe the first two points are correct, and the rest are typos. So the equation is \(y=\frac{4}{11}x+\frac{65}{11}\).

Answer:

\(y = \frac{4}{11}x+\frac{65}{11}\) (Note: There might be a typo in the table as the points do not lie on a straight line consistently)