Math Problem Statement

Identify the vertex, y-intercept, x-intercepts, and equation of symmetry from the quadratic graph.

Solution

Let's analyze the graph and solve the problems based on the quadratic equation provided.

Step-by-step analysis of the graph:

  1. Identifying the vertex: From the graph, we can see the parabola opens upwards, and the vertex is the lowest point. The x-coordinate of the vertex is at x=3x = -3 (given), and the y-coordinate looks to be at 4-4. Therefore, the vertex is: (3,4)(-3, -4)

  2. Identifying the y-intercept: The y-intercept occurs where the parabola crosses the y-axis (i.e., where x=0x = 0). From the graph, this happens at (0,2)(0, 2).

  3. Identifying the x-intercepts: The x-intercepts occur where the parabola crosses the x-axis (i.e., where y=0y = 0). From the graph, the x-intercepts appear to be at x=5x = -5 and x=1x = -1, so the intercepts are: (5,0)and(1,0)(-5, 0) \quad \text{and} \quad (-1, 0)

  4. Equation of the axis of symmetry: The axis of symmetry for a parabola passes through the vertex and is a vertical line. Since the x-coordinate of the vertex is 3-3, the equation of the axis of symmetry is: x=3x = -3

Summary:

  • Vertex: (3,4)(-3, -4)
  • Y-intercept: (0,2)(0, 2)
  • X-intercepts: (5,0)(-5, 0) and (1,0)(-1, 0)
  • Axis of symmetry: x=3x = -3

Would you like more details or have any questions on any part?

Related questions:

  1. How do you determine the vertex of a parabola algebraically?
  2. How can you derive the quadratic equation from the vertex and intercepts?
  3. What is the significance of the axis of symmetry in a quadratic function?
  4. How do you calculate the y-intercept of a parabola from its equation?
  5. What are the steps to find the x-intercepts using the quadratic formula?

Tip: For a quadratic equation in vertex form, y=a(xh)2+ky = a(x - h)^2 + k, the vertex is (h,k)(h, k), and the axis of symmetry is x=hx = h.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Graphing Parabolas
Vertex and Intercepts

Formulas

Vertex form: y = a(x - h)^2 + k
Standard form: y = ax^2 + bx + c

Theorems

Parabola Symmetry
Zeroes of a Quadratic Function

Suitable Grade Level

Grades 9-11