Question

How do you use a graph to solve an equation on the interval?

Answer 1

Hint : The given question is asked how we solve an equation by using a graph on the given interval. This question is asking the general format for solving an equation with the help of a graph. So let us explain the step by process to solve an equation on the interval only by using a graph and also with an example.

Complete step-by-step answer :
To solve an equation by using a graph on the given interval follow the steps given below:
Step 1: For a given equation on interval find the function $y$ corresponding to the equation.
Step 2: Draw the table for different values of $x$.
Step 3: Graph the function $y$, for the values on the table above.
Step 4: Observe the graph and note the points for the given equation. Those points are the solution for the given equation.
Let us consider an equation $\tan (x) = 1$ on an interval $[ - 2\pi ,2\pi ]$.
Step 1: Find the function $y$,
Given that $\tan (x) = 1$, then the function $y = \tan (x) - 1$.
Step 2: Draw the table for different values of $x$.
On the given interval assign the different values for $x$ and thus find $y = \tan (x) - 1$.
Given interval $[ - 2\pi ,2\pi ]$, the values between these interval are: $- 2\pi ,\dfrac{{ - 5\pi }}{4},\dfrac{{3\pi }}{4}$ and $2\pi$.
Now, $x = - 2\pi \Rightarrow y = \tan ( - 2\pi ) - 1$
$\tan ( - 2\pi ) = 0$, by substituting this value,
$y = 0 - 1$
$y = - 1$
$x = \dfrac{{ - 5\pi }}{4} \Rightarrow y = \tan \left( {\dfrac{{ - 5\pi }}{4}} \right) - 1$
Substituting the value $\tan \left( {\dfrac{{ - 5\pi }}{4}} \right) = - 1$,
$y = - 1 - 1$
$y = - 2$.
$x = \dfrac{{3\pi }}{4} \Rightarrow y = \tan \left( {\dfrac{{3\pi }}{4}} \right) - 1$
We know that $\tan \left( {\dfrac{{3\pi }}{4}} \right) = - 1$,
$y = - 1 - 1$
$y = - 2$.
$x = 2\pi \Rightarrow y = \tan (2\pi ) - 1$
Substitute $\tan (2\pi ) = 0$,
$y = 0 - 1$
$y = - 1$
Now draw the table for the values we found.

$x$	$y = \tan (x) - 1$	$(x,y)$
$- 2\pi$	$y = \tan ( - 2\pi ) - 1$	$( - 2\pi , - 1)$
$\dfrac{{ - 5\pi }}{4}$	$y = \tan \left( {\dfrac{{ - 5\pi }}{4}} \right) - 1$	$\left( {\dfrac{{ - 5\pi }}{4}, - 2} \right)$
$\dfrac{{3\pi }}{4}$	$y = \tan \left( {\dfrac{{3\pi }}{4}} \right) - 1$	$\left( {\dfrac{{3\pi }}{4}, - 2} \right)$
$2\pi$	$y = \tan (2\pi ) - 1$	$(2\pi , - 1)$

Step 3: Graph the function $y = \tan (x) - 1$
Draw the coordinate plane and plot the points we found for the table and connect the points.

Step 4: Now, observe the graph, the solutions for $\tan (x) = 1$ on interval $[ - 2\pi ,2\pi ]$ are $\left( {\dfrac{{ - 7\pi }}{4},\dfrac{{ - 3\pi }}{4},\dfrac{\pi }{4}\dfrac{{5\pi }}{4}} \right)$.
Hence we got the solution of $\tan (x) = 1$ on an interval $[ - 2\pi ,2\pi ]$ by using a graph.

Note : To plot a graph it is necessary to have both $x$ value and $y$ value to mark on $x$-axis and $y$- axis. Therefore convert the equation into function to obtain two points. And also be very careful while converting the equation into function. And then note the points in a form of table to identify them easily. The equation can also be solved manually by using a set of equations and formulas.