Problem 52 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{2} ;\) shift 2 units to the left and reflect in the \(x\) -axis

Step-by-Step Solution

Answer

The transformed function is \(f(x) = -(x+2)^2\).

Step 1: Shift Left

To shift the graph of the function 2 units to the left, we replace every instance of \(x\) in the function with \(x+2\). So, starting with \( f(x) = x^2 \), we get: \[ f(x) = (x+2)^2 \].

Step 2: Reflect in the x-axis

To reflect the graph in the \(x\)-axis, we multiply the entire function by \(-1\). Since our current function is \( f(x) = (x+2)^2 \), the transformation results in: \[ f(x) = -(x+2)^2 \].

Key Concepts

Quadratic Functions

Quadratic functions form the foundation of many algebraic concepts. Their general shape, called a 'parabola,' is a curve that can open upwards or downwards. The simple form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \). Here, \(a\), \(b\), and \(c\) are constants. The parameter \(a\) largely determines the direction and width of the parabola.
When \(a\) is positive, the parabola opens upwards, resembling a 'U' shape. Conversely, if \(a\) is negative, the parabola opens downwards, like an upside-down 'U'. In the simplified case where \(b\) and \(c\) are zero, the quadratic function reduces to \( f(x) = ax^2 \), which is symmetric around the y-axis.
Quadratic functions are pivotal in many fields such as physics and finance because they model various real-world scenarios, including projectile motion and area calculations.

Horizontal Shifts

A horizontal shift occurs when the graph of a function moves left or right along the x-axis. This transformation is modified by changing the input variable in the function.

• To shift a graph to the right, each instance of \(x\) in the function is replaced with \(x-h\), where \(h\) is the number of units shifted.
• To move the graph to the left, \(x\) is replaced by \(x+h\).

For example, if we start with the function \( f(x) = x^2 \) and want to shift it 2 units to the left, we substitute \(x\) with \(x+2\). This gives us the transformed function \( f(x) = (x+2)^2 \).
It’s essential to note that horizontal shifts do not alter the shape or orientation of the graph; they merely change its position along the x-axis.

Reflections

Reflections change the direction in which a graph opens, often flipping it over an axis. This transformation involves a simple manipulation with coefficients in the function equation.
Reflecting a function over the x-axis is achieved by multiplying the entire function by \(-1\). For a function \( f(x) \), the reflection over the x-axis results in \( -f(x) \). In visual terms, if the original graph opens upwards, reflecting it over the x-axis will make it open downwards.
In the case of our transformed function \( f(x) = (x+2)^2 \), reflecting it in the x-axis changes the expression to \( f(x) = -(x+2)^2 \). This effectively flips the parabola so that it opens towards the negative y-values, preserving its shape but changing its direction.