In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x² but with the given point as the vertex. (−10, −5)

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).
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 $(-10,\; -5)$ as $(h,\; k)$ in the vertex form equation. So, $h=\; -10$ and $k=\; -5$.
4. Substitute the values of $a$, $h$, and $k$ into the vertex form equation: $f(x)\; =\; 2(x\; -\; (-10))^2\; +\; (-5)$.
5. Simplify the expression inside the parentheses to get the final vertex form equation: $f(x)\; =\; 2(x\; +\; 10)^2\; -\; 5$.

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 vertex and understand the graph's transformations, such as shifts and stretches.

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. Maintaining the same 'a' preserves the shape.

Using a Given Vertex to Write the Equation

To write the equation with a specific vertex, substitute the vertex coordinates (h, k) into the vertex form f(x) = a(x - h)^2 + k. Keeping the same 'a' value ensures the parabola's shape remains unchanged, while the vertex shifts to the new point.