In Exercises 53–56, 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

1. Identify the given information: the parabola has the same shape as f(x) = 3x^2 or g(x) = -3x^2, which means the parabola opens either upward (positive leading coefficient) or downward (negative leading coefficient). Since the problem states a minimum at x = 11, the parabola must open upward, so the leading coefficient is positive, like 3 in f(x) = 3x^2.
2. Recall the vertex form of a parabola: $y=a(x-h){)}^2+k$, where (h, k) is the vertex and a determines the shape and direction of the parabola.
3. Since the minimum is 0 at x = 11, the vertex is at (11, 0). Substitute h = 11 and k = 0 into the vertex form: $y\; =\; a(x\; -\; 11)^2\; +\; 0$ or simply $y\; =\; a(x\; -\; 11)^2$.
4. Use the given shape information to determine the value of a. Because the parabola has the same shape as f(x) = 3x^2, the value of a is 3. This keeps the parabola's width and direction consistent.
5. Write the final vertex form equation using the values found: $y\; =\; 3(x\; -\; 11)^2$. This equation represents a parabola with a minimum of 0 at x = 11 and the same shape as f(x) = 3x^2.

Key Concepts

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 shape and orientation. 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 is.

Using Given Vertex Coordinates to Write the Equation

When given a vertex (h, k) and the shape of the parabola, you can write the quadratic equation by substituting these values into the vertex form. For example, if the minimum is 0 at x = 11, then h = 11 and k = 0. Using the known 'a' value from a similar parabola ensures the same shape.