Question

translation 6 units down followed by a reflection in the \(x\)-axis $$ \begin{aligned} h(x) &=f(x)-6 \\ &=2 x^2+6 x-6 \\ g(x) &=-h(x) \\ &=-\left(2 x^2+6 x-6\right) \\ &=-2 x^2-6 x+6 \end{aligned} $$

Short answer

After translating the function \(f(x) = 2x^2 + 6x\) down by 6 units, we obtain the function \(h(x) = 2x^2 + 6x - 6\). The reflection of this function in the x-axis is given by \(g(x) = -2x^2 -6x + 6\).

Step-by-step solution

Translation downwards

To translate a function \(f(x)\) downwards by 6 units, we subtract 6 from the function, like so: \(h(x) = f(x) - 6\). If \(f(x) = 2x^2 + 6x\), then after translating it down by 6 units, we have \(h(x) = 2x^2 + 6x - 6\).

Reflection in x-axis

Reflecting a function in the x-axis is equivalent to taking the negative of the function. We do this by multiplying the function by -1. Therefore, the reflected function \(g(x)\) is given by \(g(x) = -h(x) = - (2x^2 + 6x - 6) = -2x^2 -6x + 6\).

Key concepts

Translation of Functions

Function translation involves shifting a graph vertically or horizontally within the coordinate plane. Translating a quadratic function vertically means adding or subtracting a constant from the output, or "y" value, of the function.
For example, consider the function \( f(x) = 2x^2 + 6x \). To translate this function 6 units down, you subtract 6 from the entire function, making it \( h(x) = 2x^2 + 6x - 6 \).
This simple operation shifts the graph downward, without altering the shape of the parabola.

• Vertical translations move the graph up when you add a constant.
• They move it down when you subtract a constant.

Understanding translations will enhance your ability to manipulate and predict the positioning of graphs, a key skill for mastering more complex algebraic equations.

Reflection of Functions

Reflecting a function means flipping it over a particular line, most commonly the x-axis or y-axis. In the case of a reflection across the x-axis, you negate the entire output of the function.
Using the translated function \( h(x) = 2x^2 + 6x - 6 \), the reflection across the x-axis is performed by multiplying this function by \(-1\), resulting in \( g(x) = -h(x) = -2x^2 - 6x + 6 \).
This flips the graph upside down relative to its original position, changing each positive y-value to a negative one and vice versa.

• The graph remains symmetric with respect to the original parabola shape.
• It is an effective way to understand how changes in the sign of functions affect their graphs.

The ability to perform reflections is crucial in geometry and algebra, as it provides a clear visualization of function transformations.

Quadratic Functions

Quadratic functions form a fundamental class of functions often represented in the form \( ax^2 + bx + c \). These functions inherently create parabolic graphs that open upwards or downwards depending on the leading coefficient "a."
For example, the original function \( f(x) = 2x^2 + 6x \) is a quadratic function that opens upwards due to its positive leading coefficient of 2. When transformations like translations and reflections are applied, the basic parabola shape of the function remains unchanged.

• A positive "a" coefficient results in a parabola opening upwards.
• A negative "a" flips it to open downwards, especially observable after the reflection \( g(x) = -2x^2 - 6x + 6 \).

Mastering quadratic functions and their transformations builds a solid base for more intricate studies in calculus and other advanced maths topics.