Question

what is the vertical change from... what is the horizontal change from... what is the rate of change shown on the graph? the answer as a decimal rounded to the... if necessary (accompanied by a coordinate grid graph with points a and b)

Answer:

To solve the problem, we first identify the coordinates of points \( A \) and \( B \). From the graph, we can see that:

• Point \( A \) has coordinates \( (2, 2) \) (assuming the grid lines are at integer values for \( x \) and \( y \)).
• Point \( B \) has coordinates \( (4, 4) \)? Wait, no, looking again, the vertical and horizontal changes. Wait, maybe I misread. Let's check the grid. The \( x \)-axis is horizontal (right) and \( y \)-axis is vertical (up). Wait, the orange line goes from the origin? Wait, no, the points: Let's see, the dashed line from \( A \) to \( B \): horizontal change (Δx) and vertical change (Δy).

Wait, let's re-express:

1. Vertical Change (Δy): The vertical change from \( A \) to \( B \). If \( A \) is at \( (2, 2) \) and \( B \) is at \( (4, 4) \)? No, the dashed line is vertical and horizontal. Wait, the vertical dashed line: from \( A \) (y=2) to \( B \) (y=4)? No, the vertical change is \( 4 - 2 = 2 \)? Wait, no, maybe the coordinates are \( A(2, 2) \) and \( B(4, 4) \)? Wait, no, the slope (rate of change) is \( \frac{\Delta y}{\Delta x} \).

Wait, let's look at the graph again. The \( x \)-axis is labeled from 0 to 10 (horizontal), \( y \)-axis from 0 to 10 (vertical). The orange line passes through \( A \) (x=2, y=2) and \( B \) (x=4, y=4)? No, the dashed line is vertical (from y=2 to y=4) and horizontal (from x=2 to x=4). Wait, no, the vertical change (Δy) is \( 4 - 2 = 2 \), horizontal change (Δx) is \( 4 - 2 = 2 \)? But the line is decreasing? Wait, maybe I got the axes reversed. Wait, the \( x \)-axis is vertical (left to right, but labeled x) and \( y \)-axis is horizontal (bottom to top, labeled y). Wait, no, standard coordinate system: x is horizontal (right), y is vertical (up). But in the graph, the \( x \)-axis is vertical (downward) and \( y \)-axis is horizontal (rightward)? Wait, the labels: x-axis has arrow down at 10, y-axis arrow right at 10. So it's a reversed coordinate system: x increases downward, y increases rightward.

So point \( A \): x=2 (down 2), y=2 (right 2). Point \( B \): x=4 (down 4), y=4 (right 4)? No, the dashed line: vertical (x) change from 2 to 4 (so Δx = 4 - 2 = 2), horizontal (y) change from 2 to 4 (Δy = 4 - 2 = 2)? But the line is decreasing, so maybe Δy is negative. Wait, the arrow on the line is pointing downward and to the right, so as x increases (down), y increases (right)? No, that can't be. Wait, maybe the coordinates are \( A(2, 2) \) and \( B(4, 4) \) in standard (x right, y up), but the line is decreasing, so maybe I misread.

Wait, let's start over. The problem asks for vertical change, horizontal change, and rate of change (slope).

1. Vertical Change (Δy): The change in \( y \)-value from \( A \) to \( B \). If \( A \) is at \( (x_1, y_1) = (2, 2) \) and \( B \) is at \( (x_2, y_2) = (4, 4) \), then \( \Delta y = y_2 - y_1 = 4 - 2 = 2 \). But if the line is decreasing, maybe \( y_2 < y_1 \). Wait, the orange line has an arrow pointing downward and to the right, so as \( x \) increases (moves right), \( y \) decreases? Wait, no, the \( x \)-axis is vertical (down) and \( y \)-axis is horizontal (right). So \( x \) increases downward, \( y \) increases rightward. So point \( A \): x=2 (down 2), y=2 (right 2). Point \( B \): x=4 (down 4), y=4 (right 4)? No, the dashed line is vertical (x) from 2 to 4 (Δx = 4 - 2 = 2) and horizontal (y) from 2 to 4 (Δy = 4 - 2 = 2). But the line is decreasing, so maybe \( y \) decreases as \( x \) increases. Wait, maybe the coordinates are \( A(2, 4) \) and \( B(4, 2) \). Let's check:

• If \( A \) is (2, 4) and \( B \) is (4, 2):
• Vertical change (Δy) = 2 - 4 = -2
• Horizontal change (Δx) = 4 - 2 = 2
• Rate of change (slope) = \( \frac{\Delta y}{\Delta x} = \frac{-2}{2} = -1 \)

But the graph shows the dashed line from \( A \) (y=2) down to \( B \) (y=4)? No, the dashed line is vertical (x) from 2 to 4 (so x increases by 2) and horizontal (y) from 2 to 4 (y increases by 2)? But the line is decreasing, so maybe the axes are labeled incorrectly. Alternatively, maybe the vertical change is \( 4 - 2 = 2 \) (but downward, so negative), horizontal change is \( 4 - 2 = 2 \) (rightward, positive). Then rate of change is \( \frac{-2}{2} = -1 \).

Assuming the coordinates are \( A(2, 2) \) and \( B(4, 4) \) is wrong. Let's look at the grid: each square is 1 unit. The orange line goes from the top-left (x=0, y=0) down to the right. Point \( A \) is at (2, 2) (x=2, y=2), point \( B \) is at (4, 4) (x=4, y=4)? No, that would be a positive slope, but the arrow is pointing downward. Wait, maybe the \( y \)-axis is reversed: \( y \) increases leftward. So \( y=2 \) is left 2, \( y=4 \) is left 4. Then point \( A \) is (2, 2) (x=2 down, y=2 left), point \( B \) is (4, 4) (x=4 down, y=4 left). Then the line is decreasing as \( x \) increases (down) and \( y \) increases (left), so slope is \( \frac{\Delta y}{\Delta x} = \frac{4 - 2}{4 - 2} = 1 \), but that's positive. This is confusing.

Wait, the problem says "vertical change from Point A to Point B" and "horizontal change from Point A to Point B". Let's assume standard coordinates (x right, y up), but the line is decreasing. So:

• Point \( A \): (2, 4)
• Point \( B \): (4, 2)

Then:

• Vertical change (Δy) = 2 - 4 = -2
• Horizontal change (Δx) = 4 - 2 = 2
• Rate of change (slope) = \( \frac{-2}{2} = -1 \)

Alternatively, if the vertical change is the difference in \( y \)-values (regardless of direction), but usually vertical change is \( y_2 - y_1 \).

Assuming the coordinates are \( A(2, 2) \) and \( B(4, 4) \) is incorrect because the line is decreasing. So the correct coordinates must have \( y \) decreasing as \( x \) increases.

Let's check the grid again. The \( x \)-axis is vertical (downward), \( y \)-axis is horizontal (rightward). So:

• Point \( A \): x=2 (down 2), y=2 (right 2)
• Point \( B \): x=4 (down 4), y=4 (right 4)

But the line is decreasing, so as \( x \) increases (down), \( y \) decreases (left). So maybe \( y \) is leftward, so \( y=2 \) is right 2, \( y=4 \) is right 4, but the line is decreasing, so \( y \) should decrease as \( x \) increases. This is confusing. Maybe the problem has a typo, but let's proceed with the standard method.

The rate of change (slope) is \( \frac{\text{Vertical Change}}{\text{Horizontal Change}} \).

From the graph, the vertical change (Δy) is \( 4 - 2 = 2 \) (if moving up) or \( 2 - 4 = -2 \) (if moving down). The horizontal change (Δx) is \( 4 - 2 = 2 \) (if moving right).

Assuming the line is decreasing, so Δy is negative:

• Vertical Change: \( 2 - 4 = -2 \)
• Horizontal Change: \( 4 - 2 = 2 \)
• Rate of Change: \( \frac{-2}{2} = -1 \)

But maybe the vertical change is 2 (absolute value), horizontal change is 2, rate of change is -1.

Alternatively, if the coordinates are \( A(2, 2) \) and \( B(4, 4) \), then rate of change is 1, but the line is decreasing, so that's wrong.

Wait, the orange line has an arrow pointing downward and to the right, so as \( x \) increases (moves right), \( y \) decreases (moves down). So:

• Point \( A \): (2, 4)
• Point \( B \): (4, 2)

Then:

• Vertical Change (Δy) = 2 - 4 = -2
• Horizontal Change (Δx) = 4 - 2 = 2
• Rate of Change = \( \frac{-2}{2} = -1 \)

Yes, that makes sense. So:

1. Vertical Change: \( -2 \)
2. Horizontal Change: \( 2 \)
3. Rate of Change: \( -1 \)

But let's confirm with the grid. Each square is 1 unit. From \( A \) to \( B \), moving right 2 units (horizontal change +2) and down 2 units (vertical change -2). So slope is \( \frac{-2}{2} = -1 \).

So the answers are:

• Vertical Change: \( -2 \)
• Horizontal Change: \( 2 \)
• Rate of Change: \( -1 \)

But the problem might have different coordinates. Wait, the original graph: the \( x \)-axis is vertical (down) with labels 0 to 10, \( y \)-axis is horizontal (right) with labels 0 to 10. So point \( A \) is at (2, 2) (x=2 down, y=2 right), point \( B \) is at (4, 4) (x=4 down, y=4 right). But the line is decreasing, so as \( x \) increases (down), \( y \) decreases (left). So maybe \( y \) is leftward, so \( y=2 \) is left 2, \( y=4 \) is left 4. Then moving from \( A \) to \( B \), \( x \) increases by 2 (down 2), \( y \) increases by 2 (left 2), so slope is \( \frac{2}{2} = 1 \), but the line is decreasing. This is conflicting.

Alternatively, maybe the vertical change is 2 (up), horizontal change is 2 (right), but the line is decreasing, so slope is -1. I think the intended answer is:

• Vertical Change: \( -2 \)
• Horizontal Change: \( 2 \)
• Rate of Change: \( -1 \)

So the rate of change is \( -1 \).