Math Problem Statement

Use the graph of the parabola to answer the following: Does the parabola open upward or downward? Find the vertex, equation of the axis of symmetry, and the intercepts.

Solution

Here is a solution based on the given graph of the parabola:

(a) Does the parabola open upward or downward?

The parabola opens downward, as it has a maximum point (vertex) and the arms point downward.

(b) Find the coordinates of the vertex.

From the graph, the vertex is located at the highest point of the parabola: Vertex: (2, 4).

(c) Find the equation of the axis of symmetry.

The axis of symmetry is a vertical line passing through the vertex. Since the vertex's x-coordinate is x=2x = 2, the equation is: Axis of symmetry: x=2x = 2.

(d) Find the intercept(s).

  1. x-intercepts: These are the points where the parabola crosses the x-axis. From the graph, these are approximately: x-intercepts: (0, 0) and (4, 0).

  2. y-intercept: This is the point where the parabola crosses the y-axis. From the graph, this is: y-intercept: (0, 0).


If you need further clarifications or have any additional questions, feel free to ask!

Here are five follow-up questions for deeper understanding:

  1. What is the general formula for the equation of a parabola in vertex form?
  2. How do you verify the vertex coordinates algebraically?
  3. What determines whether a parabola opens upward or downward in its equation?
  4. How can you calculate the intercepts using the quadratic formula?
  5. What role does the axis of symmetry play in analyzing a parabola?

Tip: Always double-check vertex and intercept values on graphs by substituting them into the equation of the parabola.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Parabolas
Vertex
Axis of Symmetry
Intercepts

Formulas

Equation of a parabola in standard form: y = ax^2 + bx + c
Axis of symmetry: x = -b/(2a)

Theorems

Properties of parabolas in the context of quadratic functions

Suitable Grade Level

Grades 8-10