Question

use your graphing calculator to sketch the graph of the function, and then determine each of the coordinates of the x-intercepts for the function, if they exist.
$y = x^2 - 45$
sketch the graph of the function in the viewing window $-10,10 \times -50,10$. choose the correct graph below.
\\(\circ\\) a.
\\(\circ\\) b.
\\(\circ\\) c.
\\(\circ\\) d.
the x-intercepts are \\(\square\\).
(type an ordered pair. use a comma to separate answers as needed. use integers or decimals rounded to two decimal places for any numbers in the expression. type n if there is no x-intercept.)

Explanation:

Step1: Recall x - intercept definition

The x - intercepts of a function \(y = f(x)\) are the points where \(y=0\). So we set \(y = 0\) in the equation \(y=x^{2}-45\).

Step2: Solve for x

Set \(x^{2}-45 = 0\). Add 45 to both sides of the equation: \(x^{2}=45\). Then take the square root of both sides. Since if \(x^{2}=a\) (\(a\geq0\)), then \(x=\pm\sqrt{a}\), we have \(x = \pm\sqrt{45}=\pm3\sqrt{5}\approx\pm6.71\).

Step3: Write the x - intercepts as ordered pairs

The x - intercepts occur where \(y = 0\), so the ordered pairs are \((-\sqrt{45},0)\) and \((\sqrt{45},0)\) (or approximately \((-6.71, 0)\) and \((6.71, 0)\)).

Answer:

\((-6.71, 0), (6.71, 0)\) (or more precisely \((-\sqrt{45}, 0)\), \((\sqrt{45}, 0)\) which is \((-3\sqrt{5}, 0)\), \((3\sqrt{5}, 0)\) approximately \((-6.71, 0)\), \((6.71, 0)\))