Question

find the rate of change and initial value for the linear function.
the rate of change is
(simplify your answer.)

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through the points \((-6, 0)\) and \((0, -6)\)? Wait, no, let's check the axes. Wait, the x-axis and y-axis: let's find two clear points. Wait, looking at the grid, let's take two points. Let's see, when \(x = -6\), \(y = 0\)? Wait, no, maybe I got the axes reversed. Wait, the y-axis is on the right, x-axis on the bottom? Wait, the graph has x going down (since the arrow is down for x) and y going right? Wait, no, standard coordinate system: x is horizontal (left-right), y is vertical (up-down). Wait, maybe the graph is drawn with x-axis downward and y-axis to the right. Let's reorient: let's find two points. Let's take \((x_1, y_1) = (-6, 0)\) and \((x_2, y_2) = (0, -6)\)? No, that would give a negative slope, but the line is going up. Wait, maybe I misread the axes. Wait, the blue line: let's find two points. Let's see, when \(x = -6\), \(y = 0\)? No, wait, let's look at the grid. Let's take two points: let's say \((x_1, y_1) = (-6, 0)\) and \((x_2, y_2) = (0, -6)\) – no, that's a slope of \(\frac{-6 - 0}{0 - (-6)} = \frac{-6}{6} = -1\), but the line is going up. Wait, maybe the axes are reversed: x is vertical (downward) and y is horizontal (rightward). So x increases downward, y increases rightward. So let's take two points: when x (downward) is 0 (at the origin), y (rightward) is -6? No, this is confusing. Wait, maybe the correct points are \((x_1, y_1) = (0, -6)\) and \((x_2, y_2) = 6, 0\)? Wait, no, let's use the formula for slope: rate of change (slope) is \(\frac{y_2 - y_1}{x_2 - x_1}\). Let's find two points. Let's see, the line passes through \((-6, 0)\) and \((0, -6)\)? No, that's a negative slope. Wait, maybe the graph is drawn with x-axis as the vertical axis (downward) and y-axis as the horizontal axis (rightward). So x is vertical (down: positive), y is horizontal (right: positive). So let's take two points: when x = 0 (top of the x-axis), y = -6 (left on y-axis), and when x = 6 (down 6 units on x-axis), y = 0 (right 0 units? No). Wait, maybe I should look for two points where the line crosses the grid intersections. Let's see, the blue line: let's take \((x_1, y_1) = (-6, 0)\) and \((x_2, y_2) = (0, -6)\) – no, slope is -1. But the line is going up, so maybe the axes are reversed: x is upward (positive up) and y is leftward (positive left). No, this is too confusing. Wait, maybe the correct points are \((x_1, y_1) = (-6, 0)\) and \((x_2, y_2) = (0, -6)\) – no, slope is -1. Wait, the problem says "linear function", so rate of change is slope. Let's try again. Let's find two points: looking at the graph, when x = -6, y = 0; when x = 0, y = -6. Wait, no, that's a slope of -1. But the line is going up, so maybe the axes are flipped: x is horizontal (left-right, positive right), y is vertical (up-down, positive up). Then the line would be going down, but the blue line is going up. Wait, maybe the graph is mirrored. Wait, let's use the formula for slope: rate of change (slope) \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points: let's say \((x_1, y_1) = (-6, 0)\) and \((x_2, y_2) = (0, -6)\). Then \(m = \frac{-6 - 0}{0 - (-6)} = \frac{-6}{6} = -1\). Wait, but the line is going from bottom left to top right? No, in the graph, the blue line is going from the bottom left (where x is large negative, y is large negative) to top right (x large positive, y large positive). Wait, maybe I misread the axes. Let's check the grid: each square is 1 unit. Let's take two points: when x = -6, y = 0; when x = 0, y = -6 – no, that's a negative slope. Wait, maybe the correct points are \((x_1, y_1) = (0, -6)\) and \((x_2, y_2) = 6, 0\). Then \(m = \frac{0 - (-6)}{6 - 0} = \frac{6}{6} = 1\). Ah, that makes sense! So the line passes through (0, -6) and (6, 0). So x increases by 6, y increases by 6. So slope (rate of change) is \(\frac{6}{6} = 1\). Wait, but let's confirm. If x = 0, y = -6 (initial value, y-intercept). Then when x = 6, y = 0. So the change in y is 0 - (-6) = 6, change in x is 6 - 0 = 6. So slope is 6/6 = 1. So rate of change is 1.

Step2: Confirm the slope

Using the two points (0, -6) and (6, 0), the slope \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-6)}{6 - 0} = \frac{6}{6} = 1\).

Answer:

The rate of change (slope) is \(\boxed{1}\)