20. The graph of y=5x²-20x+17 in the xy-plane is shown above. If k is a constant such that the graph of y=5x²-20x+(17+k) does not intersect the x-axis, which of the following could be the value of k?
Answer: D
The value of k must be 4 for the graph to not intersect the x-axis.
For the graph of y=5x²-20x+(17+k) to not intersect the x-axis, the discriminant of the quadratic equation must be less than zero. This condition is satisfied when k is 4, resulting in a discriminant that is negative.
A) -3
If k is -3, the equation becomes y=5x²-20x+(17-3) = 5x²-20x+14. The discriminant for this equation is D = (-20)² - 4(5)(14) = 400 - 280 = 120, which is positive. Therefore, the graph would intersect the x-axis.
B) 2
With k as 2, the equation changes to y=5x²-20x+(17+2) = 5x²-20x+19. The discriminant is D = (-20)² - 4(5)(19) = 400 - 380 = 20, which is also positive. Thus, the graph intersects the x-axis in this case.
C) 3
If k is 3, the equation becomes y=5x²-20x+(17+3) = 5x²-20x+20. The discriminant is D = (-20)² - 4(5)(20) = 400 - 400 = 0. A discriminant of zero means that the graph touches the x-axis at one point, hence it does intersect the x-axis.
D) 4
When k is 4, the equation is y=5x²-20x+(17+4) = 5x²-20x+21. The discriminant is D = (-20)² - 4(5)(21) = 400 - 420 = -20, which is negative. This means the graph does not intersect the x-axis, making k=4 the correct choice.
Conclusion
The correct value of k is 4, as it ensures the discriminant is negative, indicating that the graph does not intersect the x-axis. In contrast, options A, B, and C result in positive or zero discriminants, leading to intersections with the x-axis. Therefore, only option D satisfies the condition required for the graph to remain entirely above or below the x-axis.