Graph -3/2 square root of x
The problem you're presenting seems to be asking for the graphical representation of a mathematical function. Specifically, the function in question appears to be f(x) = -3/2 * √x. The brief explanation of this question would be as follows:
You're being asked to graph a function where y, the dependent variable, is equal to negative three-halves times the square root of x, where x is the independent variable. This requires an understanding of how to graph square root functions and how to apply transformations (like vertical stretching and reflecting) to the basic parent function. In this case, the "parent function" is the square root function, √x, and the transformation involves scaling it by a factor of -3/2. The graph should take into account that the square root function is only defined for x >= 0 and that in this particular transformation, the graph will be reflected over the x-axis due to the negative sign.
$- \frac{3}{2} \sqrt{x}$
Answer
Solution:
Step:1
Determine the domain for the function $y = -\frac{3}{2}\sqrt{x}$ to identify valid $x$ values for plotting the graph.
Step:1.1
Ensure the radicand $\sqrt{x}$ is non-negative to establish the function's domain: $x \geq 0$.
Step:1.2
The function's domain is the set of $x$ values for which the function is real and defined.
Interval Notation: $[0, \infty)$ Set-Builder Notation: $\{x | x \geq 0\}$
Step:2
Identify the starting point of the graph by evaluating the function at the smallest $x$ value in the domain, which is $0$.
Step:2.1
Substitute $x$ with $0$ in the function: $f(0) = -\frac{3\sqrt{0}}{2}$.
Step:2.2
Proceed to simplify the expression.
Step:2.2.1
First, simplify the square root in the numerator.
Step:2.2.1.1
Express $0$ as a square: $f(0) = -\frac{3\sqrt{0^2}}{2}$.
Step:2.2.1.2
Extract terms from under the square root, assuming they represent positive real numbers: $f(0) = -\frac{3 \cdot 0}{2}$.
Step:2.2.2
Now, simplify the arithmetic expression.
Step:2.2.2.1
Multiply $3$ by $0$: $f(0) = -\frac{0}{2}$.
Step:2.2.2.2
Divide $0$ by $2$: $f(0) = -0$.
Step:2.2.2.3
Apply the negative sign to $0$: $f(0) = 0$.
Step:2.2.3
Conclude that the evaluated point is $0$: $(0, 0)$.
Step:3
The graph's starting point is at the coordinate $(0, 0)$.
Step:4
Choose additional $x$ values from the domain, close to the starting point, to plot more points on the graph.
Step:4.1
Evaluate the function at $x = 1$: $f(1) = -\frac{3\sqrt{1}}{2}$, resulting in the point $(1, -\frac{3}{2})$.
Step:4.1.1
Insert $x = 1$ into the function: $f(1) = -\frac{3\sqrt{1}}{2}$.
Step:4.1.2
Simplify the expression.
Step:4.1.2.1
Recognize that the square root of $1$ is $1$: $f(1) = -\frac{3 \cdot 1}{2}$.
Step:4.1.2.2
Complete the multiplication: $f(1) = -\frac{3}{2}$.
Step:4.1.2.3
The resulting value is $-\frac{3}{2}$: $y = -\frac{3}{2}$.
Step:4.2
Evaluate the function at $x = 2$: $f(2) = -\frac{3\sqrt{2}}{2}$, resulting in the point $(2, -\frac{3\sqrt{2}}{2})$.
Step:4.2.1
Insert $x = 2$ into the function: $f(2) = -\frac{3\sqrt{2}}{2}$.
Step:4.2.2
The resulting value is $-\frac{3\sqrt{2}}{2}$: $y = -\frac{3\sqrt{2}}{2}$.
Step:4.3
Utilize the points around the starting point $(0, 0)$, $(1, -1.5)$, and $(2, -2.12)$ to sketch the square root graph.
$$ \begin{array}{c|c} x & y \\ \hline 0 & 0 \\ 1 & -1.5 \\ 2 & -2.12 \\ \end{array} $$
Step:5
Plot the points on a coordinate plane and draw the graph of the function, connecting the points smoothly to illustrate the curve of the radical function.
Knowledge Notes:
To graph the function $y = -\frac{3}{2}\sqrt{x}$, one must understand several concepts:
Domain of a Function: The domain is the set of all possible input values (x-values) for which the function is defined. For square root functions, the radicand (the expression inside the square root) must be greater than or equal to zero.
Interval Notation: This is a way of writing subsets of the real number line. An interval such as $[0, \infty)$ includes all real numbers from $0$ to infinity, where $0$ is included (denoted by the square bracket) and infinity is not (denoted by the parenthesis).
Set-Builder Notation: This notation describes a set by specifying a property that its members must satisfy. For example, $\{x | x \geq 0\}$ describes all x-values that are greater than or equal to zero.
Evaluating Functions: To find points on the graph, one must substitute specific x-values into the function and calculate the corresponding y-values.
Simplifying Expressions: This involves performing arithmetic operations such as multiplication, division, and extracting square roots, where applicable.
Graphing Radical Functions: The graph of a square root function typically starts at the vertex (the lowest point on the graph for this type of function) and rises to the right. The points found by evaluating the function at specific x-values help in sketching the curve accurately.
Plotting Points: Each pair of x and y values represents a point on the coordinate plane. These points are plotted and then connected to form the graph of the function.
