The line’s unknown y-intercept, , determines whether it meets the exponential curve twice. Set the functions equal, isolate , and graph the resulting expression in Desmos to find its lowest height. Then test nearby integer heights. Don’t round the minimum down: a horizontal line below it misses the curve.
Hints
- Hint 1
At an intersection, both functions have the same output. Set their formulas equal, then subtract so the equation tells you what must equal.
- Hint 2
Graph the expression equal to and click its minimum, or lowest point. A horizontal line at height must be above that point to cross both rising sides.
- Hint 3
Because is an integer, test the whole-number heights on either side of the minimum. A height below the minimum gives no intersections, even if it’s close.
Step-by-step
Graph the heights that allow intersections
Step 1Turn an intersection into an equation for
At an intersection, the two functions have the same output. Set them equal:
Subtract to isolate :
- Step 2
Find the lowest possible height
Type , then graph . Click its minimum, the lowest point. Desmos shows about , so the smallest possible value of the left side is about . A horizontal line above this minimum can cross both rising sides; one below it cannot cross at all.
- Step 3
Rule out the integer below the minimum
Add . Desmos shows this horizontal line below the minimum, with no crossings. Any smaller integer puts the line even lower, so no gives two intersections.
- Step 4
Test the next integer height
Add . Desmos shows two distinct crossings with , near and . At each crossing, , which is the intersection equation from step 1 with . So the least integer value of that works is . Choice C.