Slope of quadratic function
A well-defined collection of distinct objects called elements or members.
Learning Outcomes
- Define the slope of a quadratic function.
- Compute the slope of any given parabola.
- Differentiate between slopes of linear equations and slopes of quadratic equations.
Exercise Questions 🤯
Hello! On this Wednesday evening here in India, I see you’re moving into some exciting new concepts! These questions are an introduction to Differential Calculus. They ask for the “slope of a parabola at a point,” which is a bit different from the slope of a straight line. Let’s cover the main idea first.
Core Concept: The Slope of a Curve (The Derivative)
Unlike a straight line which has the same slope everywhere, the steepness (slope) of a curve like a parabola is constantly changing. The “slope at a specific point” is defined as the slope of the straight line that is tangent to the curve at that exact point.
To find this slope, we use a powerful tool called the derivative. The derivative is a new function that we can calculate from the original function, and its job is to tell us the slope at any given x-value.
The Power Rule for Derivatives For functions like the ones in your questions (polynomials), we can use a simple rule called the Power Rule to find the derivative.
- If a term is $ax^n$, its derivative is $(a \cdot n)x^{n-1}$.
- The derivative of a term like $ax$ is just $a$.
- The derivative of a constant term (like +4 or +15) is 0.
We apply this rule to each term in the function to find the overall derivative, which represents the slope. Let’s use this to solve your questions.
Question 1: Slope of a Parabola
The Question: Calculate the slope of the parabola at a point $(x, y)$ obtained by plotting the following function: $y = x^2 + 2x + 4$.
Core Concept: We need to find the derivative of the function using the Power Rule. The resulting expression will be the formula for the slope at any point $x$.
Detailed Solution:
- Start with the function: $y = x^2 + 2x + 4$
- Apply the Power Rule to each term:
- For the $x^2$ term: Here, $a=1$ and $n=2$. The derivative is $(1 \cdot 2)x^{2-1} = 2x^1 = 2x$.
- For the $2x$ term: The derivative is just the coefficient, which is $2$.
- For the constant term $4$: The derivative is $0$.
- Combine the results:
- The derivative (slope function) is the sum of the individual derivatives: $2x + 2 + 0$.
Final Answer: The slope of the parabola at any point $(x, y)$ is $2x + 2$.
Question 2: Slope of a Parabola
The Question: Calculate the slope of the parabola at a point $(x, y)$ obtained by plotting the following function: $y = -5x^2 + 10x + 10$.
Core Concept: Again, we find the derivative of the function to get the general formula for its slope.
Detailed Solution:
- Start with the function: $y = -5x^2 + 10x + 10$
- Apply the Power Rule to each term:
- For the $-5x^2$ term: Here, $a=-5$ and $n=2$. The derivative is $(-5 \cdot 2)x^{2-1} = -10x^1 = -10x$.
- For the $10x$ term: The derivative is the coefficient, $10$.
- For the constant term $10$: The derivative is $0$.
- Combine the results:
- The derivative (slope function) is: $-10x + 10 + 0$.
Final Answer: The slope of the parabola at any point $(x, y)$ is $-10x + 10$.
Question 3: Slope of a Parabola at a Specific Point
The Question: Calculate the slope of the parabola given by the following function: $y = 6x^2 + 10x + 15$ at the point $(2, y)$.
Core Concept: This is a two-step process:
- First, find the derivative to get the general formula for the slope at any $x$.
- Second, substitute the specific x-value from the given point into the derivative formula to find the slope at that exact point.
Detailed Solution:
Step 1: Find the derivative (the general slope function).
- The function is $y = 6x^2 + 10x + 15$.
- Derivative of $6x^2$: $(6 \cdot 2)x^{2-1} = 12x$.
- Derivative of $10x$: $10$.
- Derivative of $15$: $0$.
- The slope at any point $x$ is given by the derivative: $Slope = 12x + 10$.
Step 2: Evaluate the slope at the specific point.
- The given point is $(2, y)$. We only need the x-coordinate, which is $x = 2$.
- Substitute $x=2$ into our slope function:
- Slope at $x=2$ is $12(2) + 10$.
- $24 + 10 = 34$.
Final Answer: The slope of the parabola at the point $(2, y)$ is 34.