Solving various equations
Parent function equations:
Solving equations: to solve an equation means to solve for x, the x-intercept, by letting y=0
- Line y=x
- Absolute Value y=IxI
- Square Root y= SQRT of x
- Cubic y=x³
- Parabola/Quadratic y=x²
- Reciprocal y=1/x
Solving equations: to solve an equation means to solve for x, the x-intercept, by letting y=0
Line:
0=2x+5
-5=2x
-5/2=x
We have just solved this equation and, by doing so, have found the x-intercept of the line.
Type y= 2x+5 into your calculator and look at the graph. This equation is a function, and it would pass the vertical line test. But, try looking at the table (or typing 2nd and then CALC then value then typing in -5/2) you should find that at -2.5, the y value is 0. Because the y value is 0, we know we have found an x-intercept. By typing the function into your calculator, you can check your work and verify that you have truly solved the function (by ensuring that there is not another x-intercept you may have missed).
- will always have 1 solution (unless a horizontal line, then no solution)
0=2x+5
-5=2x
-5/2=x
We have just solved this equation and, by doing so, have found the x-intercept of the line.
Type y= 2x+5 into your calculator and look at the graph. This equation is a function, and it would pass the vertical line test. But, try looking at the table (or typing 2nd and then CALC then value then typing in -5/2) you should find that at -2.5, the y value is 0. Because the y value is 0, we know we have found an x-intercept. By typing the function into your calculator, you can check your work and verify that you have truly solved the function (by ensuring that there is not another x-intercept you may have missed).
Absolute Value: there are three scenarios for absolute value equations:
y=I3x-4I
0=3x-4 or 0=-3x+4
4=3x or -4=-3x
4/3=x or 4/3=x
In this case, the value is the same either way, so x=4/3 is our answer. Verify the answer by checking it in your calculator. You'll see that the graph touches the x-axis only in one place.
Try another example:
y=Ix-2I+4
0=Ix-2I+4
-4=Ix-2I
-4=x-2 or -4=-x+2
-2=x or 6=x
This equation appears to have two solutions. But, always CHECK! If you type this absolute value into your calculator, you see that the graph never actually touches the x-axis so this absolute value has "no solution." But, if you hadn't checked, you would have named two solutions.
Try one more: y = I2x-3I-4. Click the button below to check your solution.
- 1 solution
- 2 solutions
- no solution
y=I3x-4I
0=3x-4 or 0=-3x+4
4=3x or -4=-3x
4/3=x or 4/3=x
In this case, the value is the same either way, so x=4/3 is our answer. Verify the answer by checking it in your calculator. You'll see that the graph touches the x-axis only in one place.
Try another example:
y=Ix-2I+4
0=Ix-2I+4
-4=Ix-2I
-4=x-2 or -4=-x+2
-2=x or 6=x
This equation appears to have two solutions. But, always CHECK! If you type this absolute value into your calculator, you see that the graph never actually touches the x-axis so this absolute value has "no solution." But, if you hadn't checked, you would have named two solutions.
Try one more: y = I2x-3I-4. Click the button below to check your solution.
Square Root: there are two possible scenarios:
y = (SQRT 23x)
0² =(SQRT 23x)²
0 = 23x
-23 = x
Check your solution. x=-23 is undetermined for this function. x=0 is the actual x-intercept and solution of this function, but I would never have know that without checking the function in a calculator.
Try: y =SQRT(15x)-4
0=SQRT(15x)-4
4²=SQRT(15x)²
16=15x
16/15=x
Our solution and x-intercept is 16/15.
- 1 solution
- no solution
y = (SQRT 23x)
0² =(SQRT 23x)²
0 = 23x
-23 = x
Check your solution. x=-23 is undetermined for this function. x=0 is the actual x-intercept and solution of this function, but I would never have know that without checking the function in a calculator.
Try: y =SQRT(15x)-4
0=SQRT(15x)-4
4²=SQRT(15x)²
16=15x
16/15=x
Our solution and x-intercept is 16/15.
Parabola/Quadratic: same three possible scenarios as absolute value functions (examine the graphs for absolute values, parabolas are very similar)
y = (x+1)²
0 = (x+1)²
SQRT(0) = SQRT(x+1)²
0 = x+1
+1 = x
Again, we have to check in our calculator to see the number of solutions. This function has 1 solution, even though we mathematically found that 1 and -1 are solutions, if we actually look at the graph, we see there is only one x-intercept at -1.
Cubic:
0 = (x+2)³
3SQRT(0) = 3SQRT(x+2)³
0 = x+2
-2 = x
x= -2 is the solution.
- 1 solution
- 2 solutions
- no solutions
y = (x+1)²
0 = (x+1)²
SQRT(0) = SQRT(x+1)²
0 = x+1
+1 = x
Again, we have to check in our calculator to see the number of solutions. This function has 1 solution, even though we mathematically found that 1 and -1 are solutions, if we actually look at the graph, we see there is only one x-intercept at -1.
Cubic:
- a cubic will have no more than 1 solution
0 = (x+2)³
3SQRT(0) = 3SQRT(x+2)³
0 = x+2
-2 = x
x= -2 is the solution.
Reciprocal:
0 = 1/(x+2)
0(x+2) = 1
0=1
No solution. If you type this graph into the calculator, you would see that the graph never touches the x-axis.
- a cubic will have no more than 1 solution
0 = 1/(x+2)
0(x+2) = 1
0=1
No solution. If you type this graph into the calculator, you would see that the graph never touches the x-axis.