The equation \( y = -4x + 10 \) defines a linear function. In this equation, \( y \) represents the dependent variable, \( x \) is the independent variable, and the equation is in the slope-intercept form, which is given by the general formula \( y = mx + b \). Here, \( m \) represents the slope and \( b \) represents the y-intercept.
Interpretation of the Components:
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Slope (\( m = -4 \)): The slope of -4 indicates that for every increase of 1 unit in \( x \), the value of \( y \) decreases by 4 units. This means the line is downward sloping.
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Y-intercept (\( b = 10 \)): The y-intercept tells us where the line crosses the y-axis. In this case, the line crosses the y-axis at \( y = 10 \) (when \( x = 0 \)).
Description of the Graph:
- The graph is a straight line that slopes downward from left to right due to the negative slope.
- It crosses the y-axis at the point (0, 10).
- To find another point on the line, you can choose an x-value, calculate the corresponding y-value, and plot the point. For example:
- If \( x = 1 \): \[ y = -4(1) + 10 = 6 \quad \Rightarrow \quad (1, 6) \]
- If \( x = 2 \): \[ y = -4(2) + 10 = 2 \quad \Rightarrow \quad (2, 2) \]
- The line continues infinitely in both directions, and the negative slope means it will keep going down as you move to the right.
In summary, the graph of \( y = -4x + 10 \) is a linear function that descends from left to right, crosses the y-axis at (0, 10), and has a slope of -4, indicating a steep decrease in \( y \) for each unit increase in \( x \).