Question

the slope of the tangent line to the parabola y = 2x^2 - 5x + 4 at the point where x = -5 is: the equation of this tangent line can be written in the form y = mx + b where m is: and where b is:

Explanation:

Step1: Find the derivative of the function

The derivative of $y = 2x^{2}-5x + 4$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $y'=4x-5$.

Step2: Calculate the slope of the tangent line

Substitute $x=-5$ into $y'$. So $m=y'|_{x = - 5}=4\times(-5)-5=-20 - 5=-25$.

Step3: Find the y - coordinate of the point on the parabola

Substitute $x = - 5$ into $y = 2x^{2}-5x + 4$. Then $y=2\times(-5)^{2}-5\times(-5)+4=2\times25 + 25+4=50+25 + 4=79$.

Step4: Find the y - intercept of the tangent line

We know the equation of the line is $y=mx + b$, and we have $m=-25$, $x=-5$, $y = 79$. Substitute these values into the equation: $79=-25\times(-5)+b$. Solving for $b$ gives $79 = 125+b$, so $b=79 - 125=-46$.

Answer:

The slope of the tangent line is $-25$.
The value of $m$ for the tangent - line equation $y=mx + b$ is $-25$.
The value of $b$ for the tangent - line equation $y=mx + b$ is $-46$.