Problem 17 - Algebra 1

Question

Sketch the graph of the function. $$y=3 \sqrt{x+1}$$

Step-by-Step Solution

Answer

The graph starts at point (-1,0) and increases on the right side forming a soft curve. It extends to the right infinitely and rises at a rate 3 times faster than the basic square root function graph due to the vertical stretch introduced by the 3 in \(3\sqrt{x+1}\).

Step 1: Understanding the basic shape

The parent function here is \(y=\sqrt{x}\). This graph starts at the origin (0, 0) and increases on the right side while forming a curve. The graph lies entirely in the first and second quadrant.

Step 2: Translating the graph

Due to \(x+1\) in our function, instead of starting from the origin, the graph will start from (-1,0) and extend to the right side. This is because the square root function is defined for \(x \geq 0\). When \(x = -1\), we get \(\sqrt{-1+1} = \sqrt{0} = 0\), so the graph will start at (-1,0).

Step 3: Vertical stretching of the graph

In our function, there is a multiplication factor of 3 outside the square root which will cause the graph to stretch vertically. That means, for a given x-value, the y-value will be 3 times the y-value on the original square root graph. This should be reflected in the graph.

Step 4: Drawing the graph

Combine all the details from above steps to sketch the graph. Start from (-1,0) and draw a curve that rises more quickly than the original square root function, reflecting the vertical stretch. The graph continuously increases for \(x \geq -1\)

Key Concepts

Square Root Function

The square root function is a basic yet essential concept in algebra. It is typically written as \(y = \sqrt{x}\). In its graph, the function starts at the origin (0,0) and extends to the right in the first quadrant. The square root function is unique because it only takes non-negative values of \(x\) due to the restriction of obtaining real numbers only. Thus, the graph lies entirely in the non-negative \(x\)-axis area. The shape is a curve that starts at the origin and gradually increases, always getting larger as \(x\) increases.

• The domain of the square root function is \(x \geq 0\).
• The range is also \(y \geq 0\) since square roots are non-negative.
• The typical graph represents a gentle curve upwards.

Understanding the parent function, like the square root, is crucial as it serves as the foundation for more complex transformations.

Vertical Stretch

Vertical stretching is a transformation that affects the y-values of a function. In simpler words, a vertical stretch makes the graph taller and narrower. For instance, if we have a function \(y = \sqrt{x}\) and transform it to \(y = a \cdot \sqrt{x}\), where \(a > 1\), the graph stretches vertically. Every point on the original graph moves upward by the factor \(a\).

• A graph \(y = 3\sqrt{x}\) stretches vertically by a factor of 3, meaning each y-value is 3 times the original value.
• This causes more rapid growth of the function's height as \(x\) grows.

Understanding vertical stretch is essential in graph transformations since it influences how rapidly the function's output increases from point to point.

Function Transformation

Function transformations adjust how graphs look compared to their parent functions. This includes shifts, stretches, and compressions along the axes. An example with the square root function is \(y = 3\sqrt{x+1}\).

• The \(+1\) inside the square root shifts the graph 1 unit to the left.
• The outside factor of 3 ensures a vertical stretch, making the graph rise faster.

Breaking down this transformation:
- First, notice the left shift due to \(x+1\), moving the starting point from \((0,0)\) to \((-1,0)\). This happens because to keep \(\sqrt{x+1}\) zero, \(x\) must be \(-1\).
- Next, the vertical stretch makes the whole graph grow taller by a constant factor.

These transformations help comprehend complex functions by visualizing minor changes to the initial graph.