Question

the function g is defined by g(x)=x² + bx, where b is a constant. if the line tangent to the graph of g at x = - 1 is parallel to the line that contains the points (0, - 2) and (2,4), what is the value of b?

Explanation:

Step1: Find the slope of the line containing (0, -2) and (2, 4)

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Substituting $x_1 = 0,y_1=-2,x_2 = 2,y_2 = 4$ gives $m=\frac{4-(-2)}{2 - 0}=\frac{6}{2}=3$.

Step2: Differentiate the function $g(x)$

Differentiate $g(x)=x^{2}+bx$ using the power - rule. The derivative $g^\prime(x)=2x + b$.

Step3: Evaluate the derivative at $x=-1$

Substitute $x = - 1$ into $g^\prime(x)$, we get $g^\prime(-1)=2(-1)+b=b - 2$.

Step4: Set the derivative equal to the slope of the line

Since the tangent line to the graph of $g$ at $x=-1$ is parallel to the line containing $(0,-2)$ and $(2,4)$, their slopes are equal. So $b - 2=3$.

Step5: Solve for $b$

Add 2 to both sides of the equation $b - 2=3$. We get $b=3 + 2=5$.

Answer:

5