Question

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] a)b)c)d)e)f) ﻿y=x−4+x+4−4y = \sqrt{x - 4} + \sqrt{x + 4} - 4
y=x−4​+x+4​−4

Accepted Answer

To find the x-intercepts of the graph, set the equation equal to zero because x-intercepts occur where the graph crosses the x-axis, meaning $$y = 0. So$$, start with the equation: $$0 = \sqrt{\,x - 4\,} + \sqrt{\,x + 4\,} - 4. $$Isolate the square root terms on one side to simplify the equation. Add 4 to both sides to get: $$4 = \sqrt{\,x - 4\,} + \sqrt{\,x + 4\,}. To $$eliminate the square roots, consider squaring both sides of the equation. Remember, when squaring, use the formula $$(a + b)^2$ = a^2$ + 2ab + b^2$. So, square both sides: 4^2$ = (\sqrt{\,x - 4\,} + \sqrt{\,x + 4\,})^2$. Expand the right side using the formula: 16 = (x - 4) + 2$$\sqrt{(x - 4)(x + 4)} + (x + 4)$$. Simplify the terms without the square root: 16 = 2x + 2$$\sqrt{(x - 4)(x + 4)}$$. Isolate the square root term: 16 - 2x = 2$$\sqrt{(x - 4)(x + 4)}$$. Then divide both sides by 2: $$\frac{16 - 2x}{2} = \sqrt{(x - 4)(x + 4)}$$. This simplifies to 8 - x$$ = \sqrt{$$x^2 - 16$$}$$. Next, square both sides again to eliminate the square root and solve for x$.

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).]
a)b)c)d)e)f)

$y = \(\sqrt{x - 4}\) + \(\sqrt{x + 4}\) - 4$

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Concetti chiave

Finding x-intercepts

Domain of Radical Functions

Graph Matching Using Intercepts

In Exercises 85–90, find the x-intercepts of the graph of each equation. Then use the x-intercepts to match the equation with its graph. [The graphs are labeled (a) through (f).] a)b)c)d)e)f) y=x−4+x+4−4y = \(\sqrt{x - 4}\) + \(\sqrt{x + 4}\) - 4 y=x−4+x+4−4