Problem 40 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=2-\sqrt{x+1} $$

Step-by-Step Solution

Answer

The graph reflects over x-axis, shifts left by 1, and up by 2 from \( y=\sqrt{x} \).

Step 1: Identify the Base Function

The given function is \( y = 2 - \sqrt{x+1} \). Start with the base function \( y = \sqrt{x} \), which is the graph of a standard square root function.

Step 2: Apply the Inside-Function Transformation

Notice the expression under the square root: \( x+1 \). This indicates a horizontal shift. For \( y = \sqrt{x+1} \), the graph of \( \sqrt{x} \) is shifted 1 unit to the left.

Step 3: Reflection and Vertical Shift

In the function \( y = 2 - \sqrt{x+1} \), the negative sign before the square root reflects the graph over the x-axis. Then, the entire graph is shifted up by 2 units due to the '2-' at the beginning of the equation.

Step 4: Sketch the Transformed Graph

Start with the graph of \( y = \sqrt{x} \). First, shift it 1 unit to the left to get \( \sqrt{x+1} \). Reflect this graph over the x-axis to get \( -\sqrt{x+1} \). Finally, shift this graph up by 2 units to obtain the graph of \( y = 2 - \sqrt{x+1} \).

Key Concepts

Graph Sketching

Graph sketching involves visualizing a function's behavior by applying known transformations to a basic graph. Instead of plotting individual points, which can be time-consuming, you apply transformations to a simple or 'parent' graph to understand the shape and position of the new function. This method helps in quickly understanding how the function behaves globally.

• You start with a basic graph, like a parabola or a square root.
• Apply transformations like shifts, stretches, or reflections sequentially.
• Each transformation alters the graph's position or shape.

By understanding each transformation, sketching becomes intuitive, allowing you to grasp the function's behavior with just a glance.

Horizontal Shift

A horizontal shift occurs when the graph of a function moves left or right. This shift happens due to a change inside the function's argument or input, such as in the expression under a square root or within a parenthesis in a polynomial.In the function \( y = 2 - \sqrt{x+1} \), the term \( x+1 \) inside the square root indicates a horizontal shift. Here, the base graph \( y = \sqrt{x} \) is shifted left by 1 unit. This might feel counterintuitive, as you might expect an addition to move it to the right. However, since we subtract this value from the input, it shifts left.

• If the transformation is \( x+c \), shift left by \(c\) units.
• If it’s \( x-c \), shift right by \(c\) units.

Such shifts are always in the opposite direction of the sign inside the function.

Reflection over x-axis

Reflection over the x-axis is a transformation that flips the graph upside down. When a function, \( f(x) \), includes a negative sign in front of it, \( -f(x) \), the graph reflects over the x-axis.In our example, the function \( y = 2 - \sqrt{x+1} \) involves a negative sign before the square root. This transforms the upward-opening graph of \( y = \sqrt{x+1} \) into a downward-opening graph, now as \( -\sqrt{x+1} \). All y-values are multiplied by -1, effectively reflecting the graph over the axis.

• This transformation inverts the graph’s y-values.
• Peaks turn to troughs, creating a mirror image below the x-axis.

Reflection helps in visually grasping how the sign impacts the direction of the graph's opening.

Vertical Shift

A vertical shift moves the graph up or down without altering its shape. This transformation occurs when you add or subtract a constant from the entire function.In the exercise, the term \( 2 - \sqrt{x+1} \) indicates a vertical shift. The addition of '2' before the negative square root shifts the entire graph upwards by 2 units.

• Addition of a constant moves the graph up.
• Subtraction shifts it down.

Vertical shifts are simple yet powerful, helping you quickly adjust the graph’s vertical position, maintaining its original shape and orientation.