Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 3x2 or g(x) = -3x2, but with the given maximum or minimum. Minimum = 0 at x = 11

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 55

Chapter 4, Problem 55

Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 3x² or g(x) = -3x², but with the given maximum or minimum. Minimum = 0 at x = 11

Verified step by step guidance

Identify the given information: the parabola has the same shape as either \(f(x) = 3x^{2}\) or \(g(x) = -3x^{2}\), and it has a minimum value of 0 at \(x = 11\).

Since the parabola has a minimum, it opens upwards, so the coefficient of \(x^{2}\) is positive. This means the shape corresponds to \(f(x) = 3x^{2}\), where the leading coefficient \(a = 3\).

Recall the vertex form of a parabola: \(y = a(x - h)^{2} + k\), where \((h, k)\) is the vertex of the parabola.

Use the vertex coordinates given: \(h = 11\) and \(k = 0\), and substitute \(a = 3\) into the vertex form to get the equation.

Write the equation as \(y = 3(x - 11)^{2} + 0\), which simplifies to \(y = 3(x - 11)^{2}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the maximum or minimum point and the parabola's shape. The value of 'a' determines the direction and width of the parabola.

Effect of the Coefficient 'a' on Parabola Shape

The coefficient 'a' in a quadratic function affects the parabola's opening direction and steepness. If 'a' is positive, the parabola opens upward with a minimum vertex; if negative, it opens downward with a maximum vertex. The absolute value of 'a' controls how narrow or wide the parabola appears.

Using Vertex Coordinates to Write the Equation

Given the vertex coordinates (h, k), you can write the quadratic equation in vertex form by substituting h and k into f(x) = a(x - h)^2 + k. This allows you to create a parabola with a specific maximum or minimum at a given point, matching the shape defined by the coefficient 'a'.

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