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. Maximum = 4 at x = -2

Verified step by step guidance

1. Identify the given information: the parabola has the same shape as either f(x) = 3x² or g(x) = -3x², and it has a maximum value of 4 at x = -2. Since the parabola has a maximum, it opens downward, so the leading coefficient should be negative, like in g(x) = -3x².
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. Use the vertex coordinates given: the vertex is at (h, k) = (-2, 4). Substitute these into the vertex form: $y\; =\; a(x\; +\; 2)^2\; +\; 4$.
4. Determine the value of $a$ by using the fact that the parabola has the same shape as $g(x)\; =\; -3x^2$. This means $a\; =\; -3$ because the shape and direction must match.
5. Write the final equation in vertex form by substituting $a\; =\; -3$ into the equation: $y\; =\; -3(x\; +\; 2)^2\; +\; 4$.

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 Leading Coefficient on Parabola Shape

The leading coefficient 'a' in a quadratic function affects the parabola's 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. Knowing the shape from a related function allows you to use the same 'a' value, adjusting only the vertex to match the given maximum or minimum.