In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x^2 but with the given point as the vertex. (5, 3)

Verified step by step guidance

1. Recall that the vertex form of a parabola is given by the equation $f\left(x\right)=a(x-h){)}^2+k$, where $(h,k)$ is the vertex of the parabola and $a$ controls the shape (width and direction) of the parabola.
2. Identify the value of $a$ from the given function $f(x)\; =\; 2x^2$. Here, $a=\; 2$, which means the parabola opens upward and is narrower than the standard parabola $y\; =\; x^2$.
3. Use the given vertex point $(5,\; 3)$ to substitute $h=\; 5$ and $k=\; 3$ into the vertex form equation, resulting in $f(x)\; =\; 2(x\; -\; 5)^2\; +\; 3$.
4. Write the final vertex form equation explicitly, ensuring it reflects the same shape as the original parabola but shifted so that the vertex is at $(5,\; 3)$.
5. Verify that the new equation maintains the same value of $a$ to keep the shape consistent, and that the vertex coordinates are correctly placed in the formula.

Key Concepts

Vertex Form of a Parabola

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 vertex and understand the parabola's position and shape.

Effect of the 'a' Coefficient on Parabola Shape

The coefficient 'a' in a quadratic function affects the parabola's width and direction. If |a| > 1, the parabola is narrower; if 0 < |a| < 1, it is wider. A positive 'a' opens upward, while a negative 'a' opens downward. Here, a = 2 means a narrower, upward-opening parabola.

Translating a Parabola to a New Vertex

To write a parabola with a given vertex, you shift the graph horizontally and vertically so the vertex moves to (h, k). Using the vertex form, replace h and k with the new vertex coordinates while keeping the same 'a' to maintain the shape.