Question

The equation of the tangent to $y=-2x^{2}+3$ at $x=1$ is

• A. $y=-4x$
• B. $y=-4x+5$
• C. $y=4x$
• D. $y=4x+5$
• E. $y=-4x+3$

Hint:

Geometry Tip: Always check your final line equation by plugging the point of tangency back into it. For $(1, 1)$, $-4(1) + 5 = 1$. It works perfectly!

Answer 1

The Correct Option is B

Concept:
The equation of a tangent line is found using the point-slope form $y - y_1 = m(x - x_1)$. The slope $m$ is the value of the first derivative of the function evaluated at the given $x$-coordinate, and $(x_1, y_1)$ is the point of tangency on the original curve.

Step 1: Find the y-coordinate of the point of tangency.
Substitute $x = 1$ into the original function to find the corresponding $y$ value: $$y_1 = -2(1)^2 + 3$$ $$y_1 = -2 + 3 = 1$$ The point of tangency is $(1, 1)$.

Step 2: Find the first derivative of the function.
Differentiate $y = -2x^2 + 3$ with respect to $x$: $$\frac{dy}{dx} = -2(2x) + 0$$ $$\frac{dy}{dx} = -4x$$

Step 3: Evaluate the derivative to find the slope (m).
Substitute $x = 1$ into the derivative to find the slope of the tangent line at that point: $$m = -4(1) = -4$$

Step 4: Set up the point-slope equation.
Use the point $(1, 1)$ and the slope $m = -4$ in the point-slope formula: $$y - y_1 = m(x - x_1)$$ $$y - 1 = -4(x - 1)$$

Step 5: Simplify into slope-intercept form.
Distribute the $-4$ and isolate $y$: $$y - 1 = -4x + 4$$ $$y = -4x + 4 + 1$$ $$y = -4x + 5$$ Hence the correct answer is (B) $y=-4x+5$.