Problem 2 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10 , find the domain and range of the given function, and draw a sketch of the graph of the function. $$ g=\left\\{(x, y) \mid y=x^{2}+2\right\\} $$

Step-by-Step Solution

Answer

Domain: \( (-\infty, +\infty) \), Range: \[ [2, +\infty) \].

Step 1: Understand the Function

The given function is in the form: \(y = x^{2} + 2\). This represents a quadratic function which is a parabola that opens upwards.

Step 2: Find the Domain

The domain of a function consists of all possible input values (x-values). For the function \(y = x^{2} + 2\), x can be any real number. Thus, the domain is \( (-\infty, +\infty) \).

Step 3: Find the Range

The range of a function consists of all possible output values (y-values). Since \(y = x^{2} + 2\) and \(x^{2}\) is always non-negative (\(x^{2} \geq 0\)), the smallest value y can take is when \(x = 0\), giving \(y = 2\). Hence, the range is \[ [2, +\infty) \].

Step 4: Draw the Graph

To sketch the graph of \(y = x^{2} + 2\), note that it is a standard parabola shifted upwards by 2 units. Plot the vertex at (0, 2) and draw a parabola opening upwards.

Key Concepts

Domain and Range of a Function

Understanding the domain and range of a function is crucial. The domain refers to all possible input values for the function, in this case, the x-values. To find the domain of the quadratic function, we need to consider where the function makes sense. For the function \(y = x^2 + 2\) , it’s evident that x can be any real number; there are no restrictions. Therefore, the domain is \((-fty, +fty)\).

On the other hand, the range refers to all possible output values, or y-values. Since \(x^2\) is always zero or positive, the smallest value of y for the function \(y = x^2 + 2\) is 2 (when \(x = 0\)). As x becomes larger or more negative, y increases without bound. That is why the range is \([2, +fty)\).

Graph of a Quadratic Function

Drawing the graph of a quadratic function involves understanding its shape and direction. The function \(y = x^2 + 2\) represents a standard parabola that opens upwards. To sketch this:

• Start by identifying the vertex of the parabola, which is the point where the function changes direction. For the function \(y = x^2 + 2\), the vertex is at (0,2).
• Next, plot the vertex on a graph at (0,2).
• Recognize the symmetery of the parabola around the y-axis.
• Draw the curve opening upwards from the vertex. The graph should show that as x moves away from the origin in either direction, y increases.

By following these steps, you’ll see that the graph accurately represents the quadratic function.

Parabola Properties

A parabola has several key properties that are useful in graphing and analysis:

• Vertex: The highest or lowest point (minimum or maximum) on the graph. For our function \(y = x^2 + 2\), the vertex is at (0, 2).
• Axis of symmetry: This is a vertical line that passes through the vertex and divides the parabola into two mirror images. For the function, it's given by x = 0.
• Direction: A parabola can open upwards or downwards. The function \(y = x^2 + 2\) opens upwards because the coefficient of x^2 is positive.
• Width: The width of a parabola is determined by the coefficient of \(x^2\). The larger the coefficient, the narrower the parabola. Here, our coefficient is 1, resulting in a standard width.

Understanding these properties will help you sketch any quadratic function easily.