Math Problem Statement

Turn into standerd form

Turn into standard form

Solution

The graph you provided shows a parabolic function f(x)f(x), and we want to express this in standard form. The parabola has the following important points:

  • The vertex is at (1,4)(1, 4).
  • Another point on the parabola is (3,4)(3, -4).

General Standard Form of a Quadratic Equation:

The standard form of a quadratic function is: f(x)=a(xh)2+kf(x) = a(x - h)^2 + k Where (h,k)(h, k) is the vertex, and aa is a constant that determines the direction and width of the parabola.

Step 1: Substitute the vertex (h,k)(h, k)

Given that the vertex is (1,4)(1, 4), we have: f(x)=a(x1)2+4f(x) = a(x - 1)^2 + 4

Step 2: Use the point (3,4)(3, -4) to find aa

Substitute x=3x = 3 and f(3)=4f(3) = -4 into the equation to solve for aa: 4=a(31)2+4-4 = a(3 - 1)^2 + 4 4=a(2)2+4-4 = a(2)^2 + 4 4=4a+4-4 = 4a + 4 44=4a-4 - 4 = 4a 8=4a-8 = 4a a=2a = -2

Step 3: Write the equation

Now that we know a=2a = -2, the equation of the parabola in vertex form is: f(x)=2(x1)2+4f(x) = -2(x - 1)^2 + 4

Step 4: Expand to Standard Form

To convert this into standard form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, expand the equation: f(x)=2(x22x+1)+4f(x) = -2(x^2 - 2x + 1) + 4 f(x)=2x2+4x2+4f(x) = -2x^2 + 4x - 2 + 4 f(x)=2x2+4x+2f(x) = -2x^2 + 4x + 2

Thus, the standard form of the quadratic equation is: f(x)=2x2+4x+2f(x) = -2x^2 + 4x + 2

Would you like more details on any part of this process?

Related Questions:

  1. How can you find the axis of symmetry from a quadratic function in standard form?
  2. How would you determine the roots of this quadratic function?
  3. Can you describe how the value of aa affects the shape of the parabola?
  4. How would the graph change if the vertex were shifted?
  5. How can you convert a standard form quadratic function back into vertex form?

Tip:

When finding the equation of a parabola, identifying the vertex and one additional point can help you quickly determine the equation.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Parabolas
Vertex Form

Formulas

f(x) = a(x - h)^2 + k
f(x) = ax^2 + bx + c

Theorems

Quadratic function
Vertex form conversion

Suitable Grade Level

Grades 9-12