Identify the initial value and rate of change for the graph shown 7- 6 5- 4 3 2 1- 0 1 2 3 4 5 6 7 8 (4 points) Initial value: 2, rate of change: 1 1 Initial value: \frac{1}{2}, rate of change: 2 1 Initial value: 2, rate of change: \frac{1}{2} Initial value: 1, rate of change: 2
The Correct Answer and Explanation is:
The correct answer is: Initial value: 2, rate of change: 1
Here is a step-by-step explanation of how to find the initial value and the rate of change from the graph.
Finding the Initial Value
The initial value of a function shown on a graph is its y-intercept. The y-intercept is the point where the line crosses the vertical y-axis. This represents the value of the function when the input, or x-value, is zero.
To find the initial value, locate the point where the purple line intersects the y-axis. Looking at the provided graph, we can see that the line passes through the y-axis at the value of 2. The coordinates of this point are (0, 2). Therefore, the initial value for the graph is 2.
Finding the Rate of Change
The rate of change for a straight line is also called its slope. The slope describes how much the y-value changes for each one-unit change in the x-value. It can be calculated by finding the “rise” (vertical change) and dividing it by the “run” (horizontal change) between any two distinct points on the line. The formula for the rate of change (slope) is (y₂ – y₁) / (x₂ – x₁).
Let’s choose two points on the line that are easy to read from the grid.
- We can use the y-intercept we already found: Point 1 = (0, 2).
- Let’s find another point where the line perfectly crosses the grid lines. For example, the point (2, 4) is clearly on the line. Let’s call this Point 2.
Now, we can calculate the rate of change:
- The rise is the change in the y-values: y₂ – y₁ = 4 – 2 = 2.
- The run is the change in the x-values: x₂ – x₁ = 2 – 0 = 2.
The rate of change is the rise divided by the run:
Rate of Change = Rise / Run = 2 / 2 = 1.
This means that for every 1 unit the line moves to the right, it also moves 1 unit up.
Conclusion
Based on our analysis, the initial value is 2 and the rate of change is 1. Matching this with the given options, the first choice is the correct answer.
