Consider the graph of the linear function [tex]$h(x)=-6+\frac{2}{3} x$[/tex]. Which quadrant will the graph not go through?
A. Quadrant I, because the slope is negative and the [tex]$y$[/tex]-intercept is positive
B. Quadrant II, because the slope is positive and the [tex]$y$[/tex]-intercept is negative
C. Quadrant III, because the slope is negative and the [tex]$y$[/tex]-intercept is positive
D. Quadrant IV, because the slope is positive and the [tex]$y$[/tex]-intercept is negative
Answer (2)
The graph of the function h ( x ) = − 6 + 3 2 x does not pass through Quadrant II. This is determined by its positive slope and negative y-intercept, which causes it to cross into Quadrants I, III, and IV, but not II. Thus, the correct choice is B.
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The linear function is h ( x ) = 3 2 x − 6 , with a positive slope and negative y -intercept.
Calculate the x -intercept by setting h ( x ) = 0 , which gives x = 9 .
The graph passes through quadrants I, III, and IV.
The graph does not pass through quadrant II, so the answer is Quadrant II .
Explanation
Analyze the function We are given the linear function h ( x ) = − 6 + f r a c 2 3 x . We want to determine which quadrant the graph of this function does not pass through. To do this, we can analyze the slope and y -intercept of the line, as well as find the x -intercept.
Find the slope and y-intercept The given function can be rewritten as h ( x ) = f r a c 2 3 x − 6 . The slope of the line is f r a c 2 3 , which is positive. The y -intercept is − 6 , which is negative. This means the line crosses the y -axis at the point ( 0 , − 6 ) .
Find the x-intercept To find the x -intercept, we set h ( x ) = 0 and solve for x : f r a c 2 3 x − 6 = 0 f r a c 2 3 x = 6 x = 6 t im es f r a c 3 2 = 9 The x -intercept is 9 , so the line crosses the x -axis at the point ( 9 , 0 ) .
Determine the quadrant the graph does not pass through Since the slope is positive and the y -intercept is negative, the graph starts from quadrant III (where both x and y are negative), passes through quadrant IV (where x is positive and y is negative), and then passes through quadrant I (where both x and y are positive). Therefore, the graph does not pass through quadrant II (where x is negative and y is positive).
Final Answer The graph of the linear function h ( x ) = − 6 + f r a c 2 3 x does not go through Quadrant II.
Examples
Linear functions are used in many real-world applications, such as modeling the cost of a service based on usage. For example, a phone plan might have a fixed monthly fee (the y-intercept) plus a charge per minute of calls (the slope). Understanding how the graph of a linear function behaves helps in predicting costs and making informed decisions. In this case, we analyzed the quadrants the line passes through, which can be useful in determining the range of possible costs or usages.
