Math Problem Statement

give me the vertex equation of this graph show all work

give me the vertex equation of this graph show all work

Solution

To find the equation of the parabola and its vertex from the provided graph, I will follow these steps:

Observing the graph:

  1. The graph is a parabola opening upwards, indicating a positive quadratic term.
  2. The vertex appears to be at the point (0,5)(0, 5), based on the symmetry and minimum value.

General Vertex Form of a Parabola:

The equation of a parabola in vertex form is:

y=a(xh)2+ky = a(x - h)^2 + k

Where:

  • (h,k)(h, k) is the vertex of the parabola.
  • aa determines the width and direction of the parabola.

From the graph, we can observe:

  • The vertex is (0,5)(0, 5), so h=0h = 0 and k=5k = 5.

Thus, the equation simplifies to:

y=a(x0)2+5ory=ax2+5y = a(x - 0)^2 + 5 \quad \text{or} \quad y = ax^2 + 5

Finding aa:

To determine the value of aa, we need another point on the parabola. From the graph, one easily identifiable point is (1,6)(1, 6), as it seems to lie on the curve.

Substituting x=1x = 1 and y=6y = 6 into the equation:

6=a(1)2+56 = a(1)^2 + 5

Solving for aa:

6=a+56 = a + 5 a=1a = 1

Final Equation:

Thus, the equation of the parabola is:

y=x2+5y = x^2 + 5

This is the equation of the parabola shown in the graph.


Would you like further details on the steps? Here are some related questions:

  1. What is the general form of a quadratic equation, and how does it relate to the vertex form?
  2. How do you determine whether a parabola opens upwards or downwards?
  3. How can you determine the vertex from a graph without the equation?
  4. What effect does changing the value of aa have on the graph of the parabola?
  5. Can this equation be converted into standard form, and how?

Tip: The coefficient aa affects the "width" or "narrowness" of the parabola—higher absolute values make it narrower, while lower values make it wider.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Parabolas
Graphing

Formulas

Vertex form of a quadratic equation: y = a(x - h)^2 + k

Theorems

Parabola Vertex Theorem

Suitable Grade Level

Grades 9-12