1.9 Graphing with Desmos: Solving Equations and Inequalities Graphically
1.9.1 Learning Objectives
By the end of this section, you should be able to:
- choose an informative Desmos viewing window;
- find zeros and intersections;
- solve equations graphically;
- solve inequalities graphically; and
- distinguish exact results from approximate results.
1.9.2 Viewing Windows
Desmos displays part of a graph inside a viewing window. A poor window can hide intercepts, intersections, turning points, or entire branches. Begin with a broad view, then zoom near the feature being studied.
Example 1.43 Choosing a Window
For \(y=x^2-25\), a window with \(-2\le x\le2\) hides both zeros. Expanding to approximately \(-7\le x\le7\) reveals the intercepts \(x=-5\) and \(x=5\).
1.9.3 Solving Equations Graphically
An equation \(f(x)=0\) can be solved graphically by graphing
\[ y=f(x) \]
and finding its \(x\)-intercepts.
Example 1.44 Solving by Zeros
Graph \(y=x^2-5x-3\) and select its intercepts. Desmos gives
\[ \boxed{x\approx-0.54\text{ and }x\approx5.54}. \]
Exercise 1.49 Use Desmos to find the zeros of \(y=x^2+x-6\).
Show answer
Answer: \(-3,2\)
Exercise 1.50 Use Desmos to estimate the real root of \(x^3+x-1=0\).
Show answer
Answer: \(0.68\)
Alternatively, \(f(x)=g(x)\) can be solved by graphing \(y=f(x)\) and \(y=g(x)\) and finding intersection \(x\)-coordinates.
Example 1.45 Solving by Intersections
To solve \(x^2=2x+3\), graph \(y=x^2\) and \(y=2x+3\). Their intersections have
\[ \boxed{x=-1\text{ and }x=3}. \]
Exercise 1.51 Use Desmos to solve \(x^2=4x+1\).
Show answer
Answer: \(2\pm\sqrt5\approx-0.24,4.24\)
Graphical solutions are usually approximations. Algebraic methods are preferred when exact answers are available, while graphs are valuable for visualizing solutions and estimating roots that are difficult to find algebraically.
1.9.4 Solving Inequalities Graphically
To solve \(f(x)>0\), locate where the graph of \(y=f(x)\) lies above the \(x\)-axis. To solve \(f(x)\le g(x)\), locate where the graph of \(f\) lies on or below the graph of \(g\).
Example 1.46 Inequality from a Graph
For \(x^2-4x-5<0\), graph \(y=x^2-4x-5\). Its zeros are \(-1\) and \(5\), and the graph lies below the axis between them:
\[ \boxed{(-1,5)}. \]
Exercise 1.52 Use Desmos to solve \(x^2-9\le0\) graphically.
Show answer
Answer: \([-3,3]\)
1.9.5 Implicit and Restricted Relations
Desmos can graph implicit relations directly, such as
\[ x^2+y^2=16, \]
and can restrict a graph to an interval using braces, such as
\[ y=x^2\{-2\le x\le3\}. \]
Example 1.47 Desmos Activity
Enter
\[ y_1=x^3-2x-2 \]
and select its \(x\)-intercept. Desmos gives the approximate real solution
\[ \boxed{x\approx1.77}. \]
Zoom in to confirm the intercept rather than relying on a distant view.
Exercise 1.53 Use Desmos to graph \(x^2+y^2=9\) and identify its intercepts.
Show answer
Answer: \((\pm3,0)\) and \((0,\pm3)\)
1.9.6 Conceptual Takeaways
- A graph only shows what lies inside its current window.
- Zeros solve \(f(x)=0\); intersections solve \(f(x)=g(x)\).
- Graphical results are usually approximate.
- Inequality solutions correspond to where one graph is above, below, or equal to another.
- Zooming and checking a broader view reduce the risk of missing solutions.
1.9.7 Skills You Should Be Able to Do
- Enter explicit, implicit, and restricted relations in Desmos.
- Adjust the viewing window.
- Identify zeros and intersections.
- Solve equations and inequalities graphically.
- Report appropriate numerical precision.
- Verify that all relevant solutions have been found.
1.9.8 Practice Problems with Solutions
Solve \(x^2-7x+10=0\).
Show Solution
The graph crosses the axis at \(\boxed{x=2,5}\).
Solve \(x^2+2=5x\).
Show Solution
Intersections of \(y=x^2+2\) and \(y=5x\) give \[ \boxed{x=\frac{5\pm\sqrt{17}}2\approx0.44,4.56}. \]
Estimate all real zeros of \(x^3-4x+1\) to two decimals.
Show Solution
The graph has three intercepts: \[ \boxed{x\approx-2.11,\ 0.25,\ 1.86}. \]
Solve \(x^2-2x-8\ge0\).
Show Solution
The zeros are \(-2\) and \(4\). The graph is on or above the axis outside them: \[ \boxed{(-\infty,-2]\cup[4,\infty)}. \]
Solve \(x^2+1<3x+5\).
Show Solution
\[ x^2-3x-4<0. \] The zeros are \(-1\) and \(4\), and the graph is below the axis between them: \[ \boxed{(-1,4)}. \]
Graph \((x-2)^2+(y+1)^2=16\) and identify its centre, radius, and intercepts.
Show Solution
The centre is \((2,-1)\) and radius is \(4\). Setting \(y=0\) gives \[ (x-2)^2=15, \] so the \(x\)-intercepts are \((2\pm\sqrt{15},0)\). Setting \(x=0\) gives \[ (y+1)^2=12, \] so the \(y\)-intercepts are \((0,-1\pm2\sqrt3)\).