Question

function a and function b are linear functions. function a (graph of a line on coordinate plane) function b (table with x and y values: x=-1,y=-9; x=6,y=19; x=7,y=23) which statement is true? the y - value of function a when x = 1 is greater than the y - value of function b when x = 1. the y - value of function a when x = 1 is less than the y - value of function b when x = 1.

Explanation:

Step1: Find the equation of Function A

From the graph, Function A passes through \((0, -2)\) (y-intercept \(b = -2\)) and \((1, 0)\) (we can see from the graph or calculate slope). The slope \(m\) of Function A: using two points \((0, -2)\) and \((1, 0)\), \(m=\frac{0 - (-2)}{1 - 0}=2\). So the equation of Function A is \(y = 2x - 2\). When \(x = 1\), \(y = 2(1)-2 = 0\).

Step2: Find the equation of Function B

Function B is a linear function. Let's use two points from the table, say \((-1, -9)\) and \((6, 19)\). The slope \(m=\frac{19 - (-9)}{6 - (-1)}=\frac{28}{7}=4\). Using point-slope form with point \((-1, -9)\): \(y - (-9)=4(x - (-1))\), so \(y + 9 = 4(x + 1)\), \(y = 4x + 4 - 9\), \(y = 4x - 5\). When \(x = 1\), \(y = 4(1)-5=-1\)? Wait, no, wait, let's check with another point. Wait, using \((6,19)\) and \((7,23)\): slope \(m=\frac{23 - 19}{7 - 6}=4\), correct. So equation is \(y = 4x - 5\)? Wait, when \(x = -1\), \(y = 4(-1)-5=-9\), correct. When \(x = 1\), \(y = 4(1)-5=-1\)? Wait, no, wait, earlier calculation for Function A at \(x = 1\) was \(0\), and Function B at \(x = 1\) is \(4(1)-5=-1\)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's recalculate Function A. Wait, from the graph, when \(x = 1\), the line passes through \((1, 0)\), and when \(x = 0\), \(y = -2\), so slope is \(\frac{0 - (-2)}{1 - 0}=2\), so equation \(y = 2x - 2\). So at \(x = 1\), \(y = 0\). Now Function B: let's take \(x = 6\), \(y = 19\); \(x = 7\), \(y = 23\). The difference in \(y\) is \(4\) when \(x\) increases by \(1\), so slope is \(4\). Let's use \(x = 6\), \(y = 19\). So equation: \(y - 19 = 4(x - 6)\), \(y = 4x - 24 + 19\), \(y = 4x - 5\). So when \(x = 1\), \(y = 4(1)-5=-1\)? Wait, but that would mean Function A at \(x = 1\) is \(0\), Function B at \(x = 1\) is \(-1\), so \(0 > -1\), so the first statement: "The y - value of Function A when \(x = 1\) is greater than the y - value of Function B when \(x = 1\)" is true? Wait, no, wait, maybe I messed up Function A's equation. Wait, looking at the graph, when \(x = 1\), the line crosses the x - axis, so \(y = 0\). Function B: let's check with \(x = 1\) in the table? Wait, the table has \(x=-1,6,7\). Wait, maybe my calculation for Function B is wrong. Wait, let's use \(x = 6\), \(y = 19\) and \(x = -1\), \(y = -9\). Slope is \(\frac{19 - (-9)}{6 - (-1)}=\frac{28}{7}=4\), correct. So equation is \(y = 4x - 5\). So at \(x = 1\), \(y = 4(1)-5=-1\). Function A at \(x = 1\) is \(0\). So \(0 > -1\), so the first statement is true? Wait, but the options are:

1. The y - value of Function A when \(x = 1\) is greater than the y - value of Function B when \(x = 1\).

1. The y - value of Function A when \(x = 1\) is less than the y - value of Function B when \(x = 1\).

So since \(0 > -1\), the first statement is true. Wait, but let's recheck Function A's equation. From the graph, when \(x = 0\), \(y = -2\), and when \(x = 1\), \(y = 0\), so slope is \(2\), equation \(y = 2x - 2\). Correct. Function B: when \(x = 1\), \(y = 4(1)-5=-1\). So Function A at \(x = 1\) is \(0\), Function B at \(x = 1\) is \(-1\). So \(0 > -1\), so the first statement is true.

Answer:

The y - value of Function A when \(x = 1\) is greater than the y - value of Function B when \(x = 1\).