Problem 6 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises 5-12, (a) identify the domain and range and (b) sketch the graph of the function. $$y=x^{2}-9$$

Step-by-Step Solution

Answer

The domain of the function \(y=x^{2}-9\) is (-∞, ∞) and the range is [-9, ∞). The graph is a parabola opening upward, crossing the x-axis at x=-3 and x=3 and the y-axis at y=-9.

Step 1: Identifying the Domain

The domain of a function includes all possible x-values. For the given function \(y=x^{2}-9\), x can be any real number because it can take any real value and the function will still produce a real y-value. Therefore, the domain is all real numbers, which can be written as (-∞, ∞).

Step 2: Identifying the Range

Now, let's determine the range of the function. Since \(y=x^{2}-9\), the minimum value of \(x^{2}\) is 0 because squares of real numbers are always non-negative. Therefore, the minimum value of \(y=x^{2}-9\) is \(0-9=-9\). As x approaches infinity, \(x^{2}\) also approaches infinity. Hence, the range is from -9 to positive infinity, this can be represented as [-9, ∞).

Step 3: Sketching the Graph

To sketch the graph, note that it is a parabola opening upwards with a vertex (which is the lowest point in this case) at y=-9. The graph intercepts the y-axis at y=-9 which corresponds to x=0. For x=\(±3\), \(y=x^{2}-9\) becomes 0, these are the points where the curve intercepts the x-axis (-3,0) and (3,0). Next, sketch the parabola with these characteristics

Key Concepts

Domain and Range of a Parabola

Understanding the domain and range of a quadratic function is essential to sketching and interpreting its graph. The domain represents all possible input values (x-values) for the function. In the function \(y = x^2 - 9\), there are no restrictions on \(x\), meaning \(x\) can be any real number. Therefore, the domain is written as \((-\infty, \infty)\). This is true for most quadratic functions where the denominator is not zero, and there are no square roots or logarithms involved.

The range illustrates all possible output values (y-values). Quadratic functions often have a minimum or maximum point due to their parabolic shape. For a function like \(y = x^2 - 9\), the lowest point on the graph is at \(-9\) since the smallest value of \(x^2\) is \(0\). Hence, the range is all values from \(-9\) to infinity, written as \([-9, \infty)\). This is because as \(x\) increases or decreases without bound, \(y\) becomes larger and larger.

Graph Sketching a Quadratic Function

Graph sketching involves understanding the shape and key features of the function. Quadratic functions graph into a shape called a parabola. In the function \(y = x^2 - 9\), the parabola opens upwards, because the \(x^2\) term has a positive coefficient.

Important points on the graph are:

• Vertex: This is the lowest point of the parabola at \((0, -9)\).
• Y-intercept: The graph cuts the y-axis where \(x = 0\), which gives \(y = -9\).
• X-intercepts: These are the points where the graph crosses the x-axis. For \(x = ±3\), \(y = 0\), giving intersection points of \((-3, 0)\) and \((3, 0)\).

With these points plotted, you can draw a smooth curve passing through them, ensuring the parabola opens upwards.

Understanding Quadratic Functions

Quadratic functions are polynomial functions of degree two. They take the form \(y = ax^2 + bx + c\). In our example, \(y = x^2 - 9\), the coefficient \(a = 1\), \(b = 0\), and \(c = -9\).

Key aspects of these functions include:

• Parabola: The graph of a quadratic function is always a parabola.
• Symmetry: The parabola is symmetric about its vertex line, known as the axis of symmetry. For our equation, this line is \(x = 0\).
• Direction: The sign of \(a\) determines if the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)). Here, since \(a = 1\), it opens upwards.

These features make quadratic functions predictable and easy to graph once you understand the basic form.