Find an equation of the tangent line to the parabola at the point .

Question

Find an equation of the tangent line to the parabola  at the point  .

Quizplus · Accepted Answer

The equation for the parabola can be written as $y=x^2+3x+2$. To find the slope of the tangent line at $(a,b)$, we take the derivative of the equation of the parabola and evaluate it at $x=a$:
$$f'(x) = 2x+3 \quad \implies \quad m = f'(a) = 2a+3.$$
We know that the point of tangency is $(a,b)=(\frac{5}{2},\frac{33}{4})$, so we have $a=\frac{5}{2}$ and $b=\frac{33}{4}$, and hence $m=2a+3=8$.

The equation of the tangent line passing through $(\frac{5}{2},\frac{33}{4})$ with slope $m=8$ can be written in point-slope form as:
$$y-\frac{33}{4} = 8(x-\frac{5}{2})$$
Simplifying gives:
$$y=8x-11$$

Therefore, the answer is B) 11eaae29_f544_5697_87fe_fbe3497b35fb_TB2067_11.

Find an Equation of the Tangent Line to the Parabola