https://file.aiquickdraw.com/mathbot/file/17275274249054hf82ian.jpg

The exercise requires finding the factorized form of second-degree functions based on given conditions.

### 1. Problem 1:
The curve of the quadratic function \( f(x) \) intercepts the x-axis at \( x = 4 \) and \( x = 5 \), and passes through the point \( (1, 24) \).

- The factorized form of a second-degree function that intercepts the x-axis at \( x = 4 \) and \( x = 5 \) is:
  \[
  f(x) = a(x - 4)(x - 5)
  \]
  Now, we need to find the constant \( a \) using the fact that \( f(1) = 24 \). Substitute \( x = 1 \) into the equation:
  \[
  24 = a(1 - 4)(1 - 5)
  \]
  \[
  24 = a(-3)(-4)
  \]
  \[
  24 = 12a
  \]
  \[
  a = 2
  \]
  Thus, the factorized form is:
  \[
  f(x) = 2(x - 4)(x - 5)
  \]

### 2. Problem 2:
The curve of the quadratic function \( f(x) \) intercepts the x-axis at \( x = -2 \), and passes through the point \( (-1, -1) \).

- The factorized form of a second-degree function that intercepts the x-axis at \( x = -2 \) can be written as:
  \[
  f(x) = a(x + 2)^2
  \]
  Since it passes through the point \( (-1, -1) \), we substitute \( x = -1 \) and \( f(x) = -1 \) into the equation to solve for \( a \):
  \[
  -1 = a(-1 + 2)^2
  \]
  \[
  -1 = a(1^2)
  \]
  \[
  -1 = a
  \]
  Thus, the factorized form is:
  \[
  f(x) = -(x + 2)^2
  \]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the general factorized form of a quadratic function with given x-intercepts \( x_1 \) and \( x_2 \)?
2. How can you determine the value of \( a \) in a quadratic function?
3. How does the sign of \( a \) affect the shape of a parabola?
4. What happens if the quadratic function only touches the x-axis at one point?
5. Can you convert the factorized form to standard form and vice versa?

**Tip**: Always use a known point (other than the x-intercepts) to solve for the coefficient \( a \) in a quadratic function.

Trouver la forme factorisée de f(x) sachant que f est une fonction du second degré dont la courbe représentative intercepte l'axe des abscisses en 4 et en 5 et passe par le point de coordonnées (1,24).

Learn how to factorize quadratic functions given specific x-intercepts and points. This exercise covers key algebraic concepts related to quadratic functions, including how to determine the constant 'a' and factor the function. 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