Problem 80 - College Algebra

Question

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-2 \sqrt{x+1}-1 $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = 2\sqrt{x+1} - 1\) is obtained by applying a vertical stretch by a factor of 2, a horizontal shift of 1 unit to the left, and a vertical shift of 1 unit downwards to the base function \(f(x) = \sqrt{x}\).

Step 1: Recognize the base function and its transformations

The base function here is \(f(x) = \sqrt{x}\), and the given function is \(g(x) = 2\sqrt{x+1} - 1\). To turn \(f(x)\) into \(g(x)\), the following transformations need to be applied: a reflection when necessary, a vertical stretch by a factor of 2, a horizontal translation of 1 unit left, and a vertical translation of 1 unit down.

Step 2: Graph the base function

Draw the graph of \(f(x) = \sqrt{x}\), which starts from the origin (0,0) and increases slowly to the right. Since square root of x is only defined for \(x \geq 0\), the graph is only present in the first quadrant of the coordinate plane.

Step 3: Apply the vertical stretch and reflection

The coefficient in front of the square root function in \(g(x)\) is 2, which introduces a vertical stretch by a factor of 2. As the coefficient is positive, no reflection over the x-axis is necessary.

Step 4: Apply the horizontal and vertical translations

The term inside the square root introduces a shift to the left by 1 unit, and the term '-1' at the end of \(g(x)\) causes a shift downwards by 1 unit. Apply these transformations to the graph.

Step 5: Finalize the graph

After applying all transformations to the graph of \(f(x)\), the result is a graph of the function \(g(x) = 2\sqrt{x+1} - 1\), which would be a graph of the square root function stretched vertically by 2, shifted downwards by 1 unit and to the left by 1 unit.

Key Concepts

Transformations of Functions

Graphing functions can be more straightforward when understanding the transformations involved. Transformations allow us to change the basic function's position, size, or orientation. For example, with the square root function, transformations can include shifts along the axes (translations), reflections, or resizing (stretches or compressions). Each transformation corresponds to a mathematical operation on the function, making it essential to distinguish between them for accurate graphing. Key transformations include:

• Translations: Shifting the graph left or right (horizontal translation) or up and down (vertical translation).
• Stretching or Compressing: Changing the graph's size vertically or horizontally.
• Reflections: Flipping the graph over an axis. While not applicable here, this is critical in other contexts.

Understanding these transformations will enable you to predict how the graph of a function will look after certain changes.

Vertical Stretch

A vertical stretch occurs when you multiply the function by a factor greater than one. This transformation makes the graph appear taller without affecting its horizontal scaling. In the function \(g(x) = 2\sqrt{x+1} - 1\), the "2" before the square root signifies a vertical stretch. To visualize this, imagine taking each point on the original function \(f(x) = \sqrt{x}\) and then moving it higher, according to the factor. The y-values are effectively doubled. As a result:

• If originally \(f(x) = \sqrt{x} = 1\), after stretching, it will become \(g(x) = 2\times1 = 2\).
• This operation preserves the x-values, so the graph keeps its shape, only stretched upwards.

The vertical stretch allows us to emphasize the changes along the y-axis while maintaining the curve's progression along the x-axis.

Horizontal Translation

Horizontal translation shifts the graph to the left or right along the x-axis. For \(g(x) = 2\sqrt{x+1} - 1\), the term \(x+1\) inside the square root moves the graph to the left by 1 unit. To decipher this, consider the "opposite direction" rule: adding inside the function translates the graph to the left, and subtracting shifts it to the right. This rule might seem counter-intuitive at first because:

• The transformation is a "subtraction" from the x-value calculation. If you needed \(\sqrt{0}\) to occur, \(x + 1\) implies that \(x\) must be \(-1\).
• Every x-value gets adjusted by 1 unit to the left, changing the graph's start point from \((0,0)\) to \((-1,0)\).

Mastering horizontal translations helps in determining how the graph has shifted along the x-axis, offering a better understanding of its position.

Vertical Translation

Vertical translation involves shifting the entire graph up or down along the y-axis. The function \(g(x) = 2\sqrt{x+1} - 1\) includes a vertical translation, represented by the "-1" outside the square root. This shifts the graph downward by 1 unit.The takeaway from this transformation is straightforward:

• Subtracting from the function's result moves it down. If the base graph of \(\sqrt{x}\) started at \((0,0)\), after translating downward by 1, it moves to \((0,-1)\).
• The translation does not affect the shape of the graph, preserving its curvature and horizontal positioning.
• All y-values decline by 1 unit, effectively shifting the entire graph lower.

Understanding vertical translations can help clarify how the function's output values are affected, giving a clearer picture of its new location.