To find the value of $a$ in the parabola $y = ax^2 + bx + c$, given the slopes at two points $(3, 2)$ and $(2, 3)$, follow these steps:

Step 1: Determine the derivative of the function.

The slope of a parabola at a point is given by the derivative: $\frac{dy}{dx} = 2ax + b$

Step 2: Use the given slopes at the points.

We are given:

• At $(3, 2)$, the slope is 37: 2a(3) + b = 37 \implies 6a + b = 37 \tag{1}
• At $(2, 3)$, the slope is 20: 2a(2) + b = 20 \implies 4a + b = 20 \tag{2}

Step 3: Solve the system of equations.

Subtract equation (2) from equation (1): $(6a + b) - (4a + b) = 37 - 20$ $2a = 17 \implies a = \frac{17}{2}$ $a = 8.5$

So, the value of $a$ is $8.5$.

Would you like to see a step-by-step explanation of the solution or any other related details?

Here are 5 related questions you might find interesting:

1. How do you find the vertex of the parabola $y = ax^2 + bx + c$?
2. What is the geometric meaning of the derivative in the context of slopes?
3. How do the coefficients $a$, $b$, and $c$ affect the shape of a parabola?
4. What is the general formula for finding the slope of a parabola at a specific point?
5. How does changing the value of $a$ affect the concavity of a parabola?

Tip: The coefficient $a$ in the equation of a parabola $y = ax^2 + bx + c$ determines the direction of the parabola's opening (upwards if $a > 0$ and downwards if $a < 0$).