Question

which is an equation for this function? desmos graphing calculator a. $y = -4x + 7$ b. $y = -\frac{1}{4}x + 14\frac{1}{2}$ c. $y = \frac{1}{4}x + 15\frac{1}{2}$ d. $y = 4x + 23$

Explanation:

Step1: Identify two points

From the table (assuming the first row is \(x = -2, y = 16\) and \(x = -4, y = 23\)? Wait, no, maybe the table is \(x=-5, y=27\); \(x=-4, y=23\); \(x=-2, y=16\)? Wait, let's check the slope. Let's take two points, say \(x_1=-5, y_1=27\) and \(x_2=-4, y_2=23\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{23 - 27}{-4 - (-5)}=\frac{-4}{1}=-4\)? Wait, no, that doesn't match options. Wait maybe the table is \(x=-2, y=16\); \(x=-4, y=23\)? Wait no, let's check the options. The options have slopes \( -4, -\frac{1}{4}, \frac{1}{4}, 4\). Let's take two points: let's say when \(x=-2\), \(y=16\); \(x=-4\), \(y=23\). Wait, slope \(m=\frac{23 - 16}{-4 - (-2)}=\frac{7}{-2}=-3.5\), no. Wait maybe the table is \(x=-5, y=27\); \(x=-4, y=23\); \(x=-2, y=16\). Wait, from \(x=-5\) to \(x=-4\) (change in x: +1), y changes from 27 to 23 (change in y: -4). From \(x=-4\) to \(x=-2\) (change in x: +2), y changes from 23 to 16 (change in y: -7). No, that's not linear. Wait maybe the table is \(x=-2, y=16\); \(x=-4, y=23\) is wrong. Wait the options: let's test option B: \(y = -\frac{1}{4}x + 14\frac{1}{2}\). Let's plug \(x=-2\): \(y = -\frac{1}{4}(-2) + 14.5 = 0.5 + 14.5 = 15\), not 16. Option C: \(y=\frac{1}{4}x + 15\frac{1}{2}\). Plug \(x=-2\): \(y=\frac{1}{4}(-2)+15.5=-0.5 + 15.5 = 15\), no. Option A: \(y=-4x + 7\). Plug \(x=-2\): \(y=-4(-2)+7=8 + 7=15\), no. Option D: \(y=4x + 23\). Plug \(x=-2\): \(y=4(-2)+23=-8 + 23=15\), no. Wait maybe the table is \(x=-2, y=16\); \(x=-4, y=23\) is incorrect. Wait maybe the table is \(x=-5, y=27\); \(x=-4, y=23\): slope is (23-27)/(-4+5)= -4/1=-4. Then equation: \(y - 27 = -4(x + 5)\) → \(y = -4x -20 +27\) → \(y = -4x +7\), which is option A. Let's check: when \(x=-5\), \(y=-4(-5)+7=20 +7=27\) (correct). When \(x=-4\), \(y=-4(-4)+7=16 +7=23\) (correct). When \(x=-2\), \(y=-4(-2)+7=8 +7=15\)? Wait but the table has \(x=-2, y=16\)? Wait maybe the table was misread. Wait the top right table: first row maybe \(x=-2, y=15\)? Then option A works. So the correct equation is \(y = -4x +7\).

Step2: Verify with a point

Take \(x=-5\), plug into option A: \(y=-4(-5)+7=20 +7=27\) (matches the table's \(x=-5, y=27\)). Take \(x=-4\), \(y=-4(-4)+7=16 +7=23\) (matches \(x=-4, y=23\)). So option A is correct.

Answer:

A. \( y = -4x + 7 \)