Problem 8 · 2017 Math Kangaroo
Medium
Algebra & Patterns
evaluate-formula
Which quadrant contains no points of the graph of the linear function \(f(x) = -3.5x + 7\)? (Quadrants are numbered I, II, III, IV anticlockwise, starting from the upper right.)
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Answer: C — III
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Hint 1 of 2
A line with negative slope and positive y-intercept rises to the upper left and falls to the lower right.
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Hint 2 of 2
Sketch where the line goes: which of the four quadrants does it simply never enter?
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Approach: trace the line by slope and intercept across the quadrants
- f(x) = -3.5x + 7 has y-intercept (0,7) and x-intercept (2,0).
- For x < 2 the line is above the x-axis (quadrants II then I); for x > 2 it drops below (quadrant IV).
- It never reaches the lower-left region, so it has no points in quadrant III.
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