What is the standard form of the parabola whose vertex form is?

The standard form of a parabola, represented as y = ax² + bx + c, can be directly derived from its vertex form, y = a(x-h)² + k, by expanding and simplifying the expression.

Understanding Parabola Forms

Parabolas are fundamental shapes in mathematics, and their equations can be expressed in different forms, each highlighting specific characteristics.

Vertex Form of a Parabola

The vertex form of a parabola is given by:

y = a(x-h)² + k

In this form:

• (h, k) represents the coordinates of the parabola's vertex, which is the lowest or highest point on the parabola.
• a determines the parabola's direction of opening and its vertical stretch or compression.
  • If a > 0, the parabola opens upwards.
  • If a < 0, the parabola opens downwards.

This form is particularly useful for quickly identifying the vertex and the axis of symmetry (x = h).

Standard Form of a Parabola

The standard form of a parabola is expressed as:

y = ax² + bx + c

In this form:

• a serves the same purpose as in the vertex form, indicating the direction and stretch.
• c represents the y-intercept of the parabola (where the parabola crosses the y-axis, when x=0).
• b influences the position of the vertex and the axis of symmetry.

The standard form is commonly used for solving quadratic equations (e.g., finding x-intercepts using the quadratic formula) and for graphing.

Converting Vertex Form to Standard Form

To convert a parabola from its vertex form y = a(x-h)² + k to its standard form y = ax² + bx + c, you need to expand the squared term and simplify the expression.

Here are the step-by-step instructions:

1. Start with the vertex form:
   y = a(x-h)² + k

2. Expand the squared binomial (x-h)²:
   Recall that (A-B)² = A² - 2AB + B². So, (x-h)² = x² - 2xh + h².
   Substitute this back into the equation:
   y = a(x² - 2xh + h²) + k

3. Distribute a to each term inside the parentheses:
   y = ax² - 2axh + ah² + k

4. Rearrange the terms to match the standard form y = ax² + bx + c:
   Group the x term and the constant terms:
   y = ax² + (-2ah)x + (ah² + k)

By comparing this result to y = ax² + bx + c, we can see the direct relationships:

• The a coefficient remains the same.
• The b coefficient in standard form is -2ah.
• The c constant in standard form is ah² + k.

Example: Converting from Vertex Form to Standard Form

Let's convert the parabola given in vertex form y = 3(x-2)² + 5 to its standard form.

Here, a = 3, h = 2, and k = 5.

1. Start with the vertex form:
   y = 3(x-2)² + 5

2. Expand (x-2)²:
   (x-2)² = x² - 2(x)(2) + 2² = x² - 4x + 4
   Substitute this back:
   y = 3(x² - 4x + 4) + 5

3. Distribute a (which is 3):
   y = 3 * x² - 3 * 4x + 3 * 4 + 5
y = 3x² - 12x + 12 + 5

4. Combine constant terms:
   y = 3x² - 12x + 17

Thus, the standard form of the parabola y = 3(x-2)² + 5 is y = 3x² - 12x + 17.