The equation of a linear function in point-slope form is [tex]$y-y_1=m\left(x-x_1\right)$[/tex]. Harold correctly wrote the equation [tex]$y=3(x-7)$[/tex] using a point and the slope. Which point did Harold use?
A. (7,3)
B. (0,7)
C. (7,0)
D. (3,7)
Answer (2)
Rewrite Harold's equation in point-slope form: y − 0 = 3 ( x − 7 ) .
Compare the equation with the general point-slope form to identify x 1 = 7 and y 1 = 0 .
The point used by Harold is ( 7 , 0 ) .
Therefore, the answer is ( 7 , 0 ) .
Explanation
Understanding the Problem The point-slope form of a linear equation is given by y − y 1 = m ( x − x 1 ) . Harold wrote the equation y = 3 ( x − 7 ) . We need to identify the point Harold used.
Rewriting the Equation Harold's equation is y = 3 ( x − 7 ) . We can rewrite it as y − 0 = 3 ( x − 7 ) .
Identifying the Point Comparing y − 0 = 3 ( x − 7 ) with the point-slope form y − y 1 = m ( x − x 1 ) , we can identify the values: y 1 = 0 , m = 3 , and x 1 = 7 . Therefore, the point used by Harold is ( 7 , 0 ) .
Examples
Understanding point-slope form is useful in various real-world scenarios. For instance, if you know the rate at which a plant grows (slope) and its height at a certain time (point), you can predict its height at any future time using the point-slope form. Similarly, in economics, if you know the rate of change of cost (slope) and the cost at a particular production level (point), you can determine the cost at any production level. This concept helps in making informed decisions and predictions based on available data.
Harold's equation in point-slope form shows that the point he used is (7, 0). Therefore, the correct answer is (7, 0). This can be confirmed by comparing the equation to the standard point-slope form and identifying the values of x 1 and y 1 .
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