H2 Maths Formulas, Techniques & Graphs >> Functions and Graphs >> Graphs >>

Solve equation by graphing in GC

Example

$$ x^4 - 2x = x^2 - 1 $$

There are two ways to solve the above equation.

Method 1: Graph the equation as 2 curves

1. In the GC, press 'y=' and plot $y_1 = x^4 - 2x$ and $y_2 = x^2 - 1$. Press 'graph' to obtain the graphs

Enter equations.png
Graph.png

2. Find the $x$-coordinates of the first point of intersection of the graphs:

  • Press '2nd' - 'trace' and select '5: intersect'
  • Select point on each curve that is close to the first point of intersection (Use < > keys to navigate and press 'enter' to select)
First point.png
Second point.png
  • Press 'enter' when prompted to 'guess'
This implies the first solution is x = 0.4257867

This implies the first solution is x = 0.4257867

3. Using the same method in Step 2, find the coordinates of the second point of intersection:

Make sure you select two points closer to the second point of intersection

Make sure you select two points closer to the second point of intersection

This implies the second solution is x = 1.3864709

This implies the second solution is x = 1.3864709

 

Method 2: Plot as a single function and find the x-intercepts

1. Bring all the terms to one side: \begin{align} x^4 - 2x & = x^2 - 1 \\ x^4 - x^2 - 2x + 1 & = 0 \end{align}

2. In the GC, press 'y=' and plot $y = x^4 - x^2 - 2x + 1$. Press 'graph' to obtain the graph

Equation.png

3. Find the first $x$-intercept of the graph:

  • Press '2nd' - 'trace' and select '2: zero'
  • Select the Left and Right bound such that the $x$-intercept is in the boundary (Use < > keys to navigate and press 'enter' to select)
Make sure the x-intercept is in the boundary!

Make sure the x-intercept is in the boundary!

  • Press 'enter' when prompted to 'guess'
This implies the first solution is x = 0.4257867

This implies the first solution is x = 0.4257867

4. Using the same method in step 3, find the second x-intercept of the graph:

This implies the second solution is x = 1.3864709

This implies the second solution is x = 1.3864709

 

Finding large values of x by accessing table of values

$$ (0.05x - 8)^5 + 5 = 0 $$

To solve the equation above by graphing, plot $y = (0.05x - 8)^5 + 5$ and find the $x$-intercepts:

No graph in standard view

No graph in standard view

No graph after zooming out

No graph after zooming out

For cases like this, we need to use the table of values by pressing ‘2nd’ - ‘graph’:

Two observations:1. The y-values are extremely small negative numbers, thus they are not plotted on the graph2. As x increases, y increases. This implies the x-intercept (i.e. y = 0) occurs for a large value of x

Two observations:

1. The y-values are extremely small negative numbers, thus they are not plotted on the graph

2. As x increases, y increases. This implies the x-intercept (i.e. y = 0) occurs for a large value of x

Press ‘2nd’ - ‘window’ and change the increment (▵Tbl) to 20. Press ‘2nd’ - ‘graph’ to access the table:

Table setup.png
With an increment of 20, we can deduce that the x-intercept (i.e. y = 0) occurs somewhere between x = 120 and x = 140

With an increment of 20, we can deduce that the x-intercept (i.e. y = 0) occurs somewhere between x = 120 and x = 140

Press ‘window’ and change

  • Xmin and Xmax to 120 and 140 respectively

  • (Optional) Ymin and Ymax to -40 and 40 respectively (to make sure y = 0 is included)

GC (Windowt).png

Press ‘graph’ and find the x-intercept (press ‘2nd’ - ‘trace’, select ‘2: zero’ and select the appropriate boundary):

This implies the solution is x = 132.4051

This implies the solution is x = 132.4051