Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the function \(g\) whose graph is the graph of \(y=\sqrt{x}\) but is stretched vertically by a factor of 7 and shifted left 5 units.

Short answer

\[ g(x) = 7 \sqrt{x + 5} \]

Step-by-step solution

Identify the base function

The base function given is \(y = \sqrt{x}\), which is a square root function.

Apply the vertical stretch

To stretch the function vertically by a factor of 7, multiply the function by 7. The new function becomes \(y = 7 \sqrt{x} \).

Apply the horizontal shift

To shift the function left by 5 units, replace \(x\) with \(x + 5\) in the function. The final function is \(y = 7 \sqrt{x + 5}\).

Key concepts

Vertical Stretching

Vertical stretching is a transformation applied to functions that increases the distance between points on the graph and the x-axis by a specific factor. This makes the graph appear taller or more narrow. In the context of our example, we are working with the square root function, which is initially defined as \( y = \sqrt{x} \). To apply a vertical stretch, you multiply the entire function by a constant factor. Here, we use 7. So, the new function becomes \( y = 7 \sqrt{x} \).

In more general terms, if you have a function \( f(x) \) and you want to apply a vertical stretch by a factor of \( k \), the transformed function will be \( y = kf(x) \). It’s important to note that if \( k \) is greater than 1, the graph stretches away from the x-axis. However, if \( 0 < k < 1 \), the graph will compress toward the x-axis. Remember, vertical stretching affects the entire function, making it taller or shorter. This is useful in various real-world applications, such as scaling outputs in graphing calculators or adjusting graphical representations in physics.

Horizontal Shifting

Horizontal shifting is another type of transformation used to move the graph of a function left or right along the x-axis. It’s done by adding or subtracting a constant to the variable inside the function. For our example, we start with the vertically stretched square root function \( y = 7 \sqrt{x} \). To shift this function to the left by 5 units, we replace \( x \) with \( x + 5 \). The resulting function is \( y = 7 \sqrt{x + 5} \).

More generally, if you have a function \( f(x) \), to shift it horizontally, you modify the input variable \( x \). To shift the function left by \( h \) units, you replace \( x \) with \( x + h \). Conversely, to shift the function right by \( h \) units, you replace \( x \) with \( x - h \). So, the general transformations would be \( y = f(x + h) \) and \( y = f(x - h) \), respectively.

This transformation is extremely useful in various applications, such as signal processing and engineering designs, where adjusting the position of a function graph is necessary without altering its shape.

Square Root Function

The square root function is a fundamental mathematical function that returns the principal square root of its input. Its general form is \( y = \sqrt{x} \). This function is defined only for non-negative values of \( x \) because the square root of a negative number is not a real number. The graph of \( y = \sqrt{x} \) starts at the origin (0,0) and increases slowly to the right.

One important property of the square root function is that it grows faster initially and then slows down as \( x \) becomes larger. It can be modified and transformed through various methods like vertical stretching, horizontal shifting, and others.

In our exercise, we began with the base function \( y = \sqrt{x} \) and applied two transformations. First, we stretched it vertically by a factor of 7 to get \( y = 7 \sqrt{x} \). Then, we shifted it 5 units to the left to get the final function \( y = 7 \sqrt{x + 5} \).

The square root function has numerous applications, including in physics for calculating velocities under constant acceleration, in computer science for algorithms requiring computational geometry, and even in finance for modeling certain types of risk.