Problem 65 - Algebra and Trigonometry

Question

In each exercise, graph the equation in a rectangular coordinate system. $$y=\frac{1}{2} x^{2}+1, x \geq 0$$

Step-by-Step Solution

Answer

The graph of equation \(y = \(\frac{1}{2} x^{2}+1\), \(x \geq 0\), is a parabola that opens upwards, starting from the y-intercept at point (0,1).

Step 1: Identify the Form of the Equation

Firstly, analyze the equation provided. The equation \(y = \(\frac{1}{2} x^{2}+1\), is a quadratic equation, and its graph will have the shape of a parabola.

Step 2: Plot Basic Points

Next, select some x-values for which to plot the respective y-values. Note that the condition here is \(x \geq 0\). Consequently, choose several positive real numbers, including zero. For each selected x-value, substitute in the equation to calculate the y-value. These (x, y) pairs are the points on the graph.

Step 3: Sketch the Graph

Last step involves sketching the graph. Begin at the point (0,1), as this is the y-intercept. Then, plot the other points obtained from Step 2. Since this is a parabola with positive x-values and a positive coefficient of \(x^2\), it opens upwards. Connect the points in a parabolic shape.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, uses two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). This system allows us to graphically represent equations and visualize their shapes.
Understanding this system is essential when graphing quadratic equations like parabolas. Each point on the graph is defined by an ordered pair \( (x, y) \). To graph an equation, plot these points on the coordinate plane, connecting them smoothly according to the equation’s form.
Some key features of this system include:

• The x-axis, which represents the horizontal component.
• The y-axis, which represents the vertical component.
• Each point denoted as \( (x,y) \), describing its position on the grid.

Parabola

A parabola is the U-shaped curve that results from graphing a quadratic equation of the form \( y = ax^2 + bx + c \). In this exercise, the equation \( y = rac{1}{2}x^2 + 1 \) represents a parabola. The coefficient \( a = \frac{1}{2} \) is positive, indicating that the parabola opens upwards.
There are several features to note about parabolas:

• Vertex: The turning point of the parabola, located at its minimum or maximum.
• Axis of Symmetry: The vertical line running through the vertex, dividing the parabola into two symmetrical halves.
• The shape is determined by the coefficient \( a \). If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.

In the given exercise, since the graph only considers \( x \geq 0 \), focus on the right half of the parabola.

Y-Intercept

The y-intercept is the point where the graph of an equation crosses the y-axis. This occurs when the value of \( x\) is zero. For the quadratic equation \( y = \frac{1}{2}x^2 + 1 \), the y-intercept is found by evaluating \( y \) at \( x = 0 \).
Substituting \( x = 0 \) into the equation yields \( y = \frac{1}{2}(0)^2 + 1 = 1 \). Hence, the graph intersects the y-axis at the point \( (0, 1) \).
The y-intercept is vital in sketching the graph, as it provides a starting point for plotting other points and forming the curve. Once identified, you can then extend the graph based on calculated points and the equation’s structure.

Positive X-Values

When graphing this specific quadratic equation, a special condition is applied: \( x \geq 0\). This means only non-negative x-values should be considered. These values are crucial for forming the right half of the parabola.
Start with small values, including zero, and increase the magnitude gradually to see how the curve rises. Calculate the corresponding \( y \)-values for selected \( x \)-values like 0, 1, 2, 3, etc.

• At \( x = 0 \), \( y = 1 \).
• At \( x = 1 \), \( y = \frac{1}{2}(1)^2 + 1 = 1.5 \).
• At \( x = 2 \), \( y = \frac{1}{2}(2)^2 + 1 = 3 \).
• Continue this to form a series of points.

By connecting these points, you'll achieve the section of the graph for the specified \( x\)-range, illustrating the upward opening of the parabola.