https://file.aiquickdraw.com/mathbot/file/1726206633229gqep55xu.jpg

This exercise set contains various questions focused on functions, including parabolas, linear equations, and solving quadratic equations. Let's break down each part:

### 1. Graph Analysis (f and p)
   You are asked to read coordinates from the graphs of \( f \) and \( p \) and solve the following:

   - **a)** Find the value of \( f(-2.5) \). 
     - From the graph of function \( f \), locate \( x = -2.5 \) and find the corresponding \( y \)-value.
   - **b)** Given \( f(0.5) = \) and \( f( \_ ) = 1.5 \), find the missing values by looking at the graph of function \( f \).
   - **c)** Find the points where \( p(1) = \_ \) and \( p(2) = \_ \) using the parabola \( p \).
   - **d)** Solve \( p(x) = 0 \), which means finding the \( x \)-intercepts of parabola \( p \).

### 2. Parabola \( p \) Equation
   - Identify the vertex of the parabola \( p \) and form its equation in vertex form: \( p(x) = a(x - h)^2 + k \).
   - Find the precise value of \( p(2.5) \) from the graph.

### 3. Line \( g \)
   - Determine the slope \( m \) and y-intercept \( b \) of the line \( g \).

### 4. Comparing Functions
   - Compare the functions \( h(x) = 2x - 3 \), \( y = 2x - 3 \), and \( a(x) = 8x - 8 \), particularly when \( x = 8 \).

### 5. Solving the Quadratic Equation
   - Solve \( x^2 - 2x = 3 \) and explain what the solutions represent in the context of the graph.

### 6. Equation \( g(x) = p(x) \)
   - Solve \( g(x) = p(x) \), noting that the solutions are \( x = 0 \) and \( x = 3 \). Discuss the significance of these solutions graphically.

### 7. Parallel Line to \( g \)
   - Find the equation of a line parallel to \( g \) with slope \( 1 \) and a negative y-intercept, ensuring it intersects the parabola at \( x = 1.5 \).

### 8. Solutions of \( f(x) = p(x) \)
   - Analyze the intersection points of \( f(x) = p(x) \), particularly \( x = -1 \), and find the linear equation corresponding to \( f(x) \).

### 9. Parabola through \( D, F, \) and \( I \)
   - Form the equation of the parabola that passes through these points and explain why this is unique.

### 10. Two Parabolas through \( D \) and \( F \)
   - Discuss how multiple parabolas can pass through the points \( D \) and \( F \).

---

Would you like detailed solutions for specific questions or further clarification? Here are five related questions to expand your understanding:

1. How do you derive the vertex form of a parabola from its graph?
2. What is the significance of the x-intercepts in solving quadratic equations graphically?
3. How can you calculate the slope of a line from a graph?
4. What happens to the solution when comparing different linear functions at specific x-values?
5. How does shifting a line parallel to itself affect its equation?

**Tip:** For solving quadratic equations graphically, always check where the graph intersects the x-axis for real solutions.

Mathematical exercises involving function graphs, parabolas, quadratic equations, and linear equations.

This exercise covers graph analysis, parabola equations, linear functions, and solving quadratic equations. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation