Determine the point on the parabola where the slope ofthe tangent is 5. Illustrate your answer with a sketch

To determine the point on a parabola where the slope of the tangent is equal to 5, we first need to know the equation of the parabola. Let's assume the standard form of the parabola is given by $y = ax^2 + bx + c$. For illustration, let’s take a typical parabolic function, such as $y = x^2$.

1.### Break Down the Problem

1. Identify the derivative of the parabola, as the derivative represents the slope of the tangent.
2. Set the derivative equal to 5 to find the x-coordinate(s) at which the tangent has this slope.
3. Substitute this x-coordinate back into the original equation to find the corresponding y-coordinate.

2.### Relevant Concepts

• The derivative of the function $y = x^2$ is computed using standard differentiation techniques: $\frac{dy}{dx} = 2x$
• Set $2x = 5$ to find when the slope of the tangent equals 5.

3.### Analysis and Detail

1. Solve the equation $2x = 5$: $x = \frac{5}{2} = 2.5$

2. Substitute $x = 2.5$ back into the original parabola to find $y$: $y = (2.5)^2 = 6.25$

4.### Verify and Summarize

• The x-coordinate where the slope of the tangent is 5 is $x = 2.5$.
• The corresponding y-coordinate is $y = 6.25$.

Final Answer

The point on the parabola $y = x^2$ where the slope of the tangent is 5 is $\left( \frac{5}{2}, 6.25 \right)$ or $(2.5, 6.25)$.

Sketch Illustration

To illustrate, you would graph the parabola $y = x^2$ and draw a tangent line at the point $(2.5, 6.25)$ which has a slope of 5. This visual representation confirms the analytical solution.