Problem 20 - Algebra 2

Question

Graph each function. State the domain and range of each function. \(y=2 \sqrt{3-4 x}+3\)

Step-by-Step Solution

Answer

Domain: \(x \leq \frac{3}{4}\). Range: \(y \geq 3\).

Step 1: Identify the Type of Function

The given function is of the form \(y = a\sqrt{b - cx} + d\), which is a square root function. This type of function typically forms a half-parabola graph.

Step 2: Determine the Domain of the Function

The expression inside the square root, \(3 - 4x\), must be non-negative for the function to be defined. Solve the inequality: \(3 - 4x \geq 0\), which simplifies to \(x \leq \frac{3}{4}\). Thus, the domain is \(x \in (-\infty, \frac{3}{4}]\).

Step 3: Determine the Range of the Function

The smallest value of the square root term \(\sqrt{3 - 4x}\) is 0, which occurs at \(x = \frac{3}{4}\). Therefore, the smallest value of \(y\) is \(2\cdot 0 + 3 = 3\). As \(x\) decreases towards \(-\infty\), the square root grows, thus \(y\) will increase without bound. Hence, the range is \(y \in [3, \infty)\).

Step 4: Graph the Function

To graph \(y = 2\sqrt{3-4x} + 3\), first plot the point where \(x = \frac{3}{4}\) corresponding to \(y = 3\). For each value of \(x < \frac{3}{4}\), compute corresponding \(y\) values to plot additional points. This results in a curve starting at \(x = \frac{3}{4}, y = 3\) and bending upwards as \(x\) decreases. Connect these points smoothly to sketch the half-parabolic shape.

Key Concepts

Domain

The domain of a function is essentially the complete set of possible input values (or 'x' values) that will make the function work and produce any real output. When dealing with square root functions, like the one given here, the expression inside the square root must be non-negative. This is because you cannot take the square root of a negative number if you want a real number result.

For the function given:

• Inside the square root we have: \(3 - 4x\).
• This expression must be greater than or equal to zero: \(3 - 4x \geq 0\).

When you solve this inequality:

• Isolate \(x\) by rearranging: \(-4x \geq -3\).
• Divide by \(-4\) (and remember to flip the inequality sign): \(x \leq \frac{3}{4}\).

So, for our function, the domain is all real numbers \(x\) less than or equal to \(\frac{3}{4}\), which is written in interval notation as \((-\infty, \frac{3}{4}]\). This means any value from negative infinity up to, and including, \(\frac{3}{4}\) is a valid input for this function.

Range

The range of a function is all the possible output values (usually 'y' values) that result from using the function. We determine the range by looking at what happens as the input (or domain) values change.

In our function:

• The term inside the square root, \(3 - 4x\), becomes zero when \(x = \frac{3}{4}\). This means the smallest value for the square root part is 0.
• Substituting for \(y\), we get: \(y = 2 \cdot 0 + 3 = 3\).

Therefore the smallest value of \(y\) is \(3\) when \(x = \frac{3}{4}\).

As \(x\) gets smaller (goes towards \(-\infty\)), \(3 - 4x\) becomes larger, resulting in the square root also increasing. Consequently,

• the function \(y = 2\sqrt{3-4x} + 3\) increases indefinitely.

So, the range is from \(3\) to infinity, denoted as \([3, \infty)\). Any value starting from \(3\) or more can be an output of this function.

Half-Parabola

In the world of graphs, a half-parabola describes a particular curve that is commonly associated with square root functions. This shape is distinctive due to its one-sided opening.

In this specific function:

• The graph starts at the point \((\frac{3}{4}, 3)\) since that is the smallest \(x\) value for which the function is defined.
• The curve extends indefinitely as \(x\) moves to the left (or decreases).

This gives it a half-parabolic shape because:

• The curve looks similar to one side of a parabola. However, unlike a full parabola, which is symmetric and opens both directions, a half-parabola only opens in one direction (in our case, upwards).

This shape can also be intuitively understood: as the negative inside of the square root decreases, the function values get larger because the square root expression grows, causing the curve to open upwards and stretch further from the starting point. The graph shows how the output grows quickly as \(x\) reduces.