Gr11 Function Quadratic

Question

A quadratic function is defined by f(x)=2x^2+5x+3. A linear function is defined by g(x)=mx+1. What value(s) of the slope of the line would make it tangent to the parabola?

Answer 1

f(x) = 2x² + 5x + 3

Slope of tangent to the parabola at (x, f(x)) = f'(x) = 4x + 5

Slope of the line = m.

So, 4x + 5 = m. Therefore, x = (m - 5) / 4.

The line and parabola must intersect at the point of tangency.

So, (m-5)/4 + 1 = 2[(m-5)/4])² + 5(m-5)/4 + 3

Multiply through by 16 to obtain: 4(m-5) + 16 = 2(m-5)² + 20(m-5) + 48

2(m² - 10m + 25) + 20m - 100 + 48 - 4m + 20 - 16 = 0

2m² - 4m +2 = 0

m² - 2m + 1 = 0

(m-1)² = 0 So, m = 1.