Math Problem Statement

vertex form when vertex is (-1/4, 3/2) and a point is (-2,0)

Solution

To find the vertex form of a quadratic equation, we use the following formula:

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

where (h,k)(h, k) is the vertex of the parabola, and aa is a constant that determines how wide or narrow the parabola is.

Given:

  • Vertex: (14,32)\left(-\frac{1}{4}, \frac{3}{2} \right)
  • A point on the parabola: (2,0)(-2, 0)

Step 1: Plug in the vertex into the vertex form.

y=a(x+14)2+32y = a\left(x + \frac{1}{4}\right)^2 + \frac{3}{2} Now, we need to find the value of aa using the point (2,0)(-2, 0).

Step 2: Substitute the point (2,0)(-2, 0) into the equation.

0=a(2+14)2+320 = a\left(-2 + \frac{1}{4}\right)^2 + \frac{3}{2}

Simplify inside the parentheses: 2+14=84+14=74-2 + \frac{1}{4} = -\frac{8}{4} + \frac{1}{4} = -\frac{7}{4} So: 0=a(74)2+320 = a\left(-\frac{7}{4}\right)^2 + \frac{3}{2}

Step 3: Simplify the equation.

(74)2=4916\left(-\frac{7}{4}\right)^2 = \frac{49}{16} 0=a4916+320 = a \cdot \frac{49}{16} + \frac{3}{2}

Step 4: Solve for aa.

First, subtract 32\frac{3}{2} from both sides: 32=a4916-\frac{3}{2} = a \cdot \frac{49}{16}

Now, multiply both sides by 1649\frac{16}{49} to isolate aa: a=321649=4898=2449a = -\frac{3}{2} \cdot \frac{16}{49} = -\frac{48}{98} = -\frac{24}{49}

Step 5: Write the final equation.

Now that we know a=2449a = -\frac{24}{49}, the vertex form of the equation is:

y=2449(x+14)2+32y = -\frac{24}{49}\left(x + \frac{1}{4}\right)^2 + \frac{3}{2}


Do you need further details or clarification?

Here are five related questions:

  1. How do you convert vertex form to standard form?
  2. How do you determine the direction of a parabola using aa?
  3. What does the vertex of a quadratic function represent?
  4. How can you find the x-intercepts from vertex form?
  5. How do you complete the square to transform a standard form equation into vertex form?

Tip: The sign of aa determines whether the parabola opens upwards (a>0a > 0) or downwards (a<0a < 0).

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Vertex Form of a Parabola

Formulas

y = a(x - h)^2 + k (Vertex Form of a Quadratic Equation)

Theorems

Quadratic Functions Theorem (relating the vertex form to the standard form)

Suitable Grade Level

Grades 10-12