Question

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

Short 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-by-step solution

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 (-∞, ∞).

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, ∞).

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