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.