Math Problem Statement

Solve and the vertex is -3 1

What is the standard form of the quadratic function y = ax^2 + bx + c shown in the graph below?

Solution

To determine the standard form of the quadratic function y=ax2+bx+cy = ax^2 + bx + c given the vertex and the values for aa, bb, and cc, we can use the vertex form and convert it to standard form.

Given Data:

  • Vertex: (3,1)(-3, 1)
  • Values: a=13a = \frac{1}{3}, b=2b = 2, c=4c = 4

Step-by-Step Solution:

  1. Vertex Form: The vertex form of a quadratic equation is: y=a(xh)2+ky = a(x - h)^2 + k where (h,k)(h, k) is the vertex. Here, h=3h = -3 and k=1k = 1.

    Substituting a=13a = \frac{1}{3}, h=3h = -3, and k=1k = 1: y=13(x+3)2+1y = \frac{1}{3}(x + 3)^2 + 1

  2. Expanding to Standard Form: Now, expand (x+3)2(x + 3)^2 and simplify to get the standard form y=ax2+bx+cy = ax^2 + bx + c.

    y=13(x2+6x+9)+1y = \frac{1}{3}(x^2 + 6x + 9) + 1 y=13x2+2x+3+1y = \frac{1}{3}x^2 + 2x + 3 + 1 y=13x2+2x+4y = \frac{1}{3}x^2 + 2x + 4

  3. Conclusion: The standard form of the function is: y=13x2+2x+4y = \frac{1}{3}x^2 + 2x + 4

So, the values in the question match the equation derived here.

Would you like further explanation on the expansion or on identifying parts of the vertex form?

Related Questions:

  1. How do you convert from vertex form to standard form for other values of aa, hh, and kk?
  2. What does the value of aa indicate about the parabola's direction and width?
  3. How can you find the vertex from the standard form of a quadratic equation?
  4. What role does the value of bb play in determining the shape of the parabola?
  5. How do you complete the square to convert from standard form to vertex form?

Tip:

Always double-check each term when expanding squares to avoid small errors, as these can affect the entire equation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Graphing

Formulas

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

Theorems

Expansion of quadratic expressions

Suitable Grade Level

Grades 9-11