Problem 27 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ h(x)=-\sqrt{x}+3 $$

Step-by-Step Solution

Answer

Reflect and shift the square root graph up by 3 units, starting from (0, 3).

Step 1: Identify the Basic Function

The basic function here is \( y = \sqrt{x} \), which is the square root function. Its graph is a curve that starts at the origin and increases, moving to the right.

Step 2: Reflect the Function

The function given is \( h(x) = -\sqrt{x} + 3 \). The \(-\sqrt{x}\) part indicates a reflection over the x-axis. This flips the standard square root function downwards.

Step 3: Translate the Function

The +3 in \( -\sqrt{x} + 3 \) shifts the graph of \(-\sqrt{x}\) vertically upward by 3 units. Originally, \( \sqrt{x} \) starts at the origin, so the translated graph will start at (0, 3).

Step 4: Sketch the Graph

Plot the function by starting at point (0, 3). From there, draw a curve that moves left to right that begins at (0,3) and decreases as it moves outward, indicating the negative square root. The reflected and shifted nature means it will never go below the line \( y=3 \) as it approaches infinity to the right.

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental function in mathematics denoted by \( y = \sqrt{x} \). Its graph is characterized by its distinct curve shape, which starts from the origin point (0,0) and extends infinitely to the right in the first quadrant. This function increases as you move along the x-axis. Its domain is all non-negative values of x (i.e., from 0 to infinity), and its range is also non-negative values, since the square root of a number can never be negative.

• The graph crosses the origin (0, 0).
• It rises slowly and continuously as x increases.
• The shape resembles half a parabola.

Recognizing this basic shape is critical when graphing transformations.

Exploring Reflection Over the X-Axis

Reflection over the x-axis is a transformation that flips a graph upside down. For a function \( f(x) \), reflecting it over the x-axis results in \( -f(x) \). This transformation changes the direction in which the graph appears.
In the context of our exercise, the function \( -\sqrt{x} \) is the result of reflecting the basic square root function across the x-axis. The curve starts at the origin, but instead of going upward, it extends downward.

• Reflection across the x-axis changes the sign of the function’s y-values.
• For \( y = -\sqrt{x} \), every point (x, y) on the \( \sqrt{x} \) graph becomes (x, -y) on the \( -\sqrt{x} \) graph.
• The graph doesn’t change location horizontally, only vertically in direction.

Navigating Vertical Translation

Vertical translation involves shifting a graph up or down without altering its shape or orientation. This transformation is performed by adding or subtracting a constant to a function.
In the equation \( h(x) = -\sqrt{x} + 3 \), the "+3" indicates a vertical translation 3 units upward from where the reflection graph of \( -\sqrt{x} \) originally lies. This means every y-value on the graph is increased by 3.

• Vertical translation doesn't affect the function's domain or range.
• The graph of \( -\sqrt{x} \) is lifted upwards, starting from (0, 3).
• This translates all points upwards but maintains their horizontal position.

Understanding vertical translation emphasizes how functions can be moved vertically in a systematic way without changing their inherent properties.

Introduction to Function Transformation

Function transformation includes any operation that alters the original position, shape, or orientation of a function's graph. These transformations include reflections, translations, stretches, and compressions.
In the example of \( h(x) = -\sqrt{x} + 3 \), we are dealing with two key transformations:

• Reflection: Flipping the \( \sqrt{x} \) graph over the x-axis to get \( -\sqrt{x} \).
• Vertical Translation: Moving the resulted graph three units upwards.

These transformations can be applied in sequence to alter the graph's layout significantly. Knowing how these transformations influence a graph helps in efficiently sketching graphs of more complex functions. It's like peeling layers, where each layer reveals a slight alteration to the previous function.