Math Problem Statement

Find the x- and y-intercepts from the graph of the equation y = 3x - x^2.

Solution

From the graph and the equation y=3xx2y = 3x - x^2, let's determine the intercepts.

Finding the x-intercepts:

The x-intercepts occur where y=0y = 0. Setting the equation to zero: 0=3xx20 = 3x - x^2 x23x=0x^2 - 3x = 0 Factoring out xx: x(x3)=0x(x - 3) = 0 Setting each factor equal to zero: x=0andx=3x = 0 \quad \text{and} \quad x = 3

Finding the y-intercept:

The y-intercept occurs where x=0x = 0. Substituting into the equation: y=3(0)02=0y = 3(0) - 0^2 = 0

Solutions:

  • x-intercept (smaller x-value): (0,0)(0, 0)
  • x-intercept (larger x-value): (3,0)(3, 0)
  • y-intercept: (0,0)(0, 0)

These answers are consistent with the graph provided. If you have any questions or need further details, feel free to ask.

Related Questions:

  1. How do you determine the vertex of the parabola y=3xx2y = 3x - x^2?
  2. What is the axis of symmetry for the equation y=3xx2y = 3x - x^2?
  3. How can you verify the intercepts algebraically without a graph?
  4. What transformations have been applied if the original equation was y=x2y = -x^2?
  5. How would the graph change if the equation was y=x23xy = x^2 - 3x?

Tip:

When you have a quadratic equation in the form y=ax2+bx+cy = ax^2 + bx + c, setting y=0y = 0 and factoring can directly give you the x-intercepts.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Intercepts
Factoring

Formulas

Quadratic equation: y = ax^2 + bx + c
Intercepts: Solve for x when y = 0 (for x-intercepts), solve for y when x = 0 (for y-intercept)

Theorems

Factoring method to solve quadratic equations

Suitable Grade Level

Grades 8-10