The initial value of a linear function is found by substituting 0 for x in the equation and solving for y. This y-value represents the y-intercept, which is the initial value.
Here's a breakdown of the process:
Understanding the Initial Value
The initial value of a linear function represents the value of the function (i.e., the y-value) when the independent variable x is zero. In a real-world context, it often represents the starting point or baseline value of a quantity. Think of it as what happens before anything else has happened.
The Linear Equation
A linear function can be represented by the equation:
y = mx + b
Where:
- y is the dependent variable
- x is the independent variable
- m is the slope (rate of change)
- b is the y-intercept (initial value)
Steps to Find the Initial Value
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Identify the Equation: Know the equation of your linear function (y = mx + b). If you have two points, you can determine the equation using slope-intercept form.
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Substitute 0 for x: Replace the x variable in the equation with the value 0.
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Solve for y: Simplify the equation and solve for y. The resulting value of y is the initial value.
Example
Let's say you have the linear function:
y = 2x + 3
To find the initial value:
- Substitute x = 0: y = 2(0) + 3
- Simplify: y = 0 + 3
- Solve for y: y = 3
Therefore, the initial value of the function y = 2x + 3 is 3.
Why This Works
Setting x = 0 effectively eliminates the slope (m) term from the equation (mx becomes zero), leaving only the y-intercept (b). Since y = mx + b, then y = m(0) + b, which simplifies to y = b. The y-intercept, b, is the initial value.
Finding Initial Value from a Graph
If you have the graph of a linear function, the initial value is simply the y-coordinate of the point where the line intersects the y-axis.
