Equations and 2.3 Inequalities...

 Graphing Calculators; Solving Equations and Inequalities Graphically 2.3

Solving Equations and Inequalities Graphically

  To do this, we must first draw a graph using a graphing device, this is your TI-83/84 calculator.

  You should know how to use it to graph several equations at once   Y=, WINDOW, ZOOM, TRACE, GRAPH buttons   You should also know the second function of each of these buttons,

particularly FORMAT, CALC and TABLE

Viewing Rectangle

  A graphing calculator or computer displays a rectangular portion of the graph of an equation in a display window or viewing screen.

  We call this a viewing rectangle, find it by selecting WINDOW

Viewing Rectangle

 Let’s choose:

  The x-values to range from a minimum value of Xmin = a to a maximum value of Xmax = b

  The y-values to range from a minimum value of Ymin = c to a maximum value of Ymax = d.

Viewing Rectangle   Then, the displayed portion of the graph

lies in the rectangle [a, b] x [c, d] = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}

  We refer to this as the [a, b] by [c, d] viewing rectangle.

Two Graphs on the Same Screen

  Graph the equations y = 3x2 – 6x + 1 and y = 0.23x – 2.25

together in the viewing rectangle [1, –3] by [–2.5, 1.5]

  Enter on of these above as y1 and the other as y2

  Press the GRAPH button to see both of these equations

  Do the graphs intersect in this viewing rectangle?


 The figure shows the essential features of both graphs.

  One is a parabola and the other is a line.

  It looks as if the graphs intersect near the point (1, –2).


  Use the ZOOM button to ZOOM in or create a ZOOMbox around the point to see what is really happening

Using a Graphing Calculator

 Most graphing calculators can only graph equations in which y is isolated on one side of the equal sign.

  The next example shows how to graph equations that don’t have this property.

Graphing a Circle

 Graph the circle x2 + y2 = 1.

  We first solve for y—to isolate it on one side of the equal sign.

y2 = 1 – x2 (Subtract x2) y = ± (Take square roots)

Graphing a Circle

 Thus, the circle is described by the graphs of two equations:

  The first equation represents the top half of the circle (because y ≥ 0).

  The second represents the bottom half (y ≤ 0).

Graphing a Circle

  If we graph the first equation in the viewing rectangle [–2, 2] by [–2, 2], we get the semicircle shown.

  The graph of the second equation is the semicircle shown.

  Graphing these semicircles together on the same viewing screen, we get the full circle shown.

 Solving Equations Graphically

Solving Equations Algebraically   In Chapter 1, we learned how to solve equations.

  To solve an equation like 3x – 5 = 0, we used the algebraic method.

  This means we used the rules of algebra to isolate x on one side of the equation.

Solving Equations Algebraically   We view x as an unknown and we use

the rules of algebra to hunt it down.

  Here are the steps: 3x – 5 = 0

3x = 5 (Add 5)

x = 5/3 (Divide by 3)

Solving Equations Graphically

 We can also solve this equation by the graphical method.

  We view x as a variable and sketch the graph of the equation y = 3x – 5.

  Different values for x give different values for y.

  Our goal is to find the value of x for which y = 0.


 From the graph, we see that y = 0 when x ≈ 1.7.

  The solution is x ≈ 1.7.


 We summarize these methods here.

Solving a Quadratic Equation

 Solve the quadratic equations algebraically and graphically.

(a)  x2 – 4x + 2 = 0 (b)  x2 – 4x + 4 = 0 (c)  x2 – 4x + 6 = 0

Solving Algebraically

  There are two solutions:


  There is just one solution, x = 2.


  There is no real solution.

Solving Graphically   Now, see the power of graphing the equations as

simultaneous equations in your calculator   y = x2 – 4x + 2

y = x2 – 4x + 4 y = x2 – 4x + 6

  By determining the x-intercepts of the graphs, we find the following solutions. In our calculator, this is found when you do the CALC function which the second TRACE

Solving Graphically

 x ≈ 0.6 and x ≈ 3.4

Example (a)

Solving Quadratic Equations Graphically

  There is no x-intercept for that last one, you can see it does not cross the x-axis therefore the equation has no solution.

  The graphs in Figure 6 show visually why a quadratic equation may have two solutions, one solution, or no real solution.


  In the next example, we use the graphical method to solve an equation that is extremely difficult to solve algebraically.

E.g. 6—Solving an Equation in an Interval

 Solve the equation in the interval [1, 6].

  We need to find all solutions x that satisfy 1 ≤ x ≤ 6.

  So, we will graph the equation in a viewing rectangle for which the x-values are restricted to this interval.

Solving an Equation in an Interval

 The figure shows the graph of the equation in the viewing rectangle [1, 6] by [–5, 5].

  There are two x-intercepts in this rectangle.


  To find the x-intercepts on your calculator   Use CALC, select 2 for zero   Put the cursor before and after each point and put your guess

around the intercept…Do you see why you need to do this twice?

1 2


 Zooming in, we see that the solutions are: x ≈ 2.128 and x ≈ 3.72

 Solving Inequalities Graphically

Solving Inequalities Graphically

 To solve the inequality graphically, we draw the graph of y = x2 – 5x + 6.

  Our goal is to find those values of x for which y ≤ 0.

Solving Inequalities Graphically

 These are simply the x-values for which the graph lies below the x-axis.

  We see that the solution of the inequality is the interval [2, 3].

Solving an Inequality Graphically

 Solve the inequality

3.7x2 + 1.3x – 1.9 ≤ 2.0 – 1.4x  Ouch, hard algebraically but pretty easy on the calculator graphically

Solving an Inequality Graphically   We graph the equations

y1 = 3.7x2 + 1.3x – 1.9 y2 = 2.0 – 1.4x

in the same viewing rectangle.

Solving an Inequality Graphically

 We are interested in those values of x for which y1 ≤ y2.

  The solution are points for which the graph of y2 (blue) lies on or above the graph of y1 (red)

  You could pick a point to test, maybe the origin?? You try it.

E.g. 9—Solving an Inequality Graphically

 Solve the inequality x3 – 5x2 ≥ –8

  We write the inequality as:

x3 – 5x2 + 8 ≥ 0

E.g. 9—Solving an Inequality Graphically

 Then, we graph the equation y = x3 – 5x2 + 8 in the viewing rectangle [–6, 6] by [–15, 15]

  The solution consists of those intervals on which the graph lies on or above the x-axis.

Equations and 2.3 Inequalities...

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