If the slope of parabola y=ax2+bx+c*y*=*ax*2+*bx*+*c*, where a,b,c∈*a*,*b*,*c*∈RR \{0}{0} at points (3,2)(3,2) and (2,3)(2,3) are 3737 and 2020 respectively, then find the value of a*a*.

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 \( a < 0 \)).

If the slope of parabola y = ax^2 + bx + c, where a, b, c ∈ R \ {0} at points (3, 2) and (2, 3) are 37 and 20 respectively, then find the value of a.

Learn how to find the coefficient 'a' in the quadratic equation y = ax^2 + bx + c by using the slopes at two given points. This step-by-step solution covers key algebraic and calculus concepts, including derivatives and systems of equations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation