Graph of a Linear Function |
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The linear function can be written as f(x) = mx + b in function form or y = mx + b in equation form, where the parameters m and b are real constants and x is a real variable. The constant m is often called the slope of the line or gradient, while b gives the value of the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.
Changing these parameters affects their graphs. For example the equation y = x + 2, where m = 1 and b = 2 has the graph |
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| Suppose we change the value of m into m = 2. Our equation becomes y = 2x + 2 and has the graph |
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As you can see, the line y = 2x +2 is steeper than the line y = x + 2.
Now, suppose we change the value of b from the first equation into b = 5, we have the equation y = x + 5. The graph of this linear equation is |
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From this graph, we can say that the line y = x + 5 translates the line y = x + 2 by 3 steps upward.
Hence, changing m makes the line steeper or shallower, while changing b moves the line up or down.
We now know that in the case of a linear function y = f(x), expressed in slope-intercept form y = mx + b, m and b are parameters. Also, f(x) = kx represents a direct variation (proportional relationship). |
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Functions Involving Absolute Value |
From the definition of the absolute value of a number, the equation y = |x| is given by
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| Sketching the graph of the equation y = |x|, we have |
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| Notice that y = |x| is a combination of two linear equations: |
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(1) y = x, where m = 1 and b = 0, and
(2) y = -x, where m = -1 and b = 0.
Since these equations are both linear, we follow the same rule as what we have learned about changing the parameters m and b of a linear equation.
Changing m would make the lines steeper or shallower, while changing b would translate the lines upward or downward. For example,
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| The graph of this equation is |
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Try these questions |
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Which equation does this graph represent? |
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| a. |
y = |x+3|+3 |
| b. |
y = |x|+3 |
| c. |
y = |x+3| |
| d. |
y = |x - 3| - 3 |
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What is the difference between the graph of y=|x + 3| +3 and the graph of y =|x|? Recall that the graph of y = |x| is |
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| a. |
The graph of y = |x| is steeper than that of y = |x+3|+3. |
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The graph of y = |x+3|+3 is same as the graph of y = |x|. |
| c. |
The graph of y=|x| is translates the graph of y = |x+3|+3 by 3 units to the left and 3 units upward. |
| d. |
The graph of y=|x+3|+3 translates the graph of y =|x| by 3 units to the left and 3 units upward. |
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3. |
Complete the statement: In a linear equation y = mx + b, |
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| a. |
the parameter b is equal to the y-intercept |
| b. |
the parameter m is equal to the slope of the line |
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when the parameter m increases, the slope of the line becomes steeper |
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All of the above. |
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ANSWERS TO PRACTICE TEST QUESTIONS |
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- A. y = |x+3|+3.
- D. The graph of y=|x+3|+3 translates the graph of y =|x| by 3 units to the left and 3 units upward.
- D. All of the above. All statements are true about the linear equation as discussed in the lesson.
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