The advent of graphing calculators and scientific calculators has made advanced math much simpler to work with, but the best calculator in the world won’t do you much good unless you understand what you’re doing in the first place.
Any student, technician or engineer who’s working through higher math problems with a scientific calculator needs to have a firm grasp of the theory and principle behind the problems.
One area where these devices help hugely is in calculating quadratic equations and finding X intercepts.
Before Using an X Intercept Calculator, Know the Quadratic Equation
Researchers and archaeologists have found quadratic equation-type problems that were being worked on as far back as 2000 B.C. Let’s go back to advanced algebra days. Algebra defines a quadratic equation as any equation with the form:
ax^2 + bx + c = 0
In this type of equation, x represents the unknown, with a, b, and c representing known numbers (with a not equal to 0). If a = 0, then the equation is a linear equation rather than quadratic. The numbers a, b, and c are the equation’s coefficients; they can be defined, respectively, as the quadratic coefficient, linear coefficient and constant (or free term).
A quadratic equation that includes only one unknown is called “univariate,” and a quadratic equation that only contains powers of x as non-negative integers is known as a “polynomial equation.” If the greatest power is two, it’s a second-degree polynomial equation.
There are actually a few different approaches for resolving quadratic equations.
Quadratic equations are solved by factoring, by completing the square, using a quadratic formula or by graphing. Quadratic equations with either real or complex coefficients can have two solutions, called roots. These two solutions may or may not be real, and may or may not be distinct.
A quadratic equation of ax^2 + bx + c = 0 can also be expressed as a product of (px + q)(rx + s) = 0. In a case like this, inspection can determine the values of p, q, r, and s in a way that shows the two forms as equivalent to each other. If the quadratic equation is expressed in that second form, then “Zero Factor Property” will show that the quadratic equation is resolved if px + 1 = 0 or rx + s = 0.
In most cases, factoring by inspection is a preferred method for resolving quadratic equations. In the case of x^2 + bx + c = 0, the factorization would include the form (x + 1) (x + s), and the numbers represented by q and s would add up to b, with a final product of c.
Using a Graphing Calculator To Find X and Y Intercepts
Now that we have some ground rules on the theory behind quadratic equations, let’s talk a little about how to use a graphing calculator as an x intercept calculator. The great thing about these devices is that they feature built-in tools that enable you to work through these instances and find the intercepts without actually doing the algebra.
- Enter the equation, then press the Y= button on the device
- Clear any existing equations from the device
- Be sure to pay careful attention and include any parentheses and operators in the equation
- Graph the equation and press the ZOOM button, being sure to use a zoom that’s compatible with your equation
- Zoom will need to include proper x and y intercepts
- After finding the y intercept, press the TRACE button, followed by the 0 button
- The cursor should now be at the y intercept, with x = 0
- At the bottom of the screen, you should now see the y intercept
- Next, find the x intercept(s) by pressing the 2ND key followed by the CALC key to access the trace menu
- Scroll down to ZERO and press ENTER
- Use the arrow keys to scroll to the left of the x intercept, then press ENTER
- Next, scroll to the right of the intercept and press ENTER twice to display the x intercept at the bottom of the screen
- Remember that some equations may have more than one x intercept – you’ll need to perform these steps for each one
Using a Calculator To Solve Quadratic Equations
Remembering that a quadratic equation is one that can be expressed as ax^2 + bx + c = 0, here’s how to use a calculator to work your way through this mainstay of algebra and advanced math.
- Check the equation, ensuring it’s written in the standard form of ax^2 + bx + c =0. If not, rewrite the equation to conform
- Press MODE button repeatedly and eventually EQN will appear on the screen
- Press 1 and enter the calculation mode
- Press the right arrow, followed by 2 – this should enable you to start a quadratic equation
- Enter the values of a, b, and c by entering each number and pressing the – sign
- Scroll down and see the solutions to the equation
Determining the X Intercept of a Function
Let’s think about definitions for a moment. We all know that the x-axis is the horizontal axis of a graph, and the y-axis is the vertical. An x-intercept, then, is the point on a line that crosses the x-axis of a graph and represents a function.
The x-intercept is expressed as (x,0) since the y-coordinate will always be zero at the x-intercept. So, if we already know the y-intercept of a function and the slope, the x-intercept can be calculated using the formula (y -b)/m = x, with m equal to slope, y equals zero and b equal to y-intercept.
- In the equation (y – b)/m=x, substitute the slope for m and the y-intercept for x
- Example: if slope equals 5 and the y-intercept is 3, express the formula as (y – 3)/5 = x
- Substitute 0 for y in this equation, since y is zero at the x-intercept
- Per the previous example, the equation would be (0 -3)/5=x
- Now, you can solve the equation to determine the value of x. Per the previous example, (0 – 3)/5 = x, solve the numerator first
- Subtract 0 from 3 and the result will be -3/5 = x
- Divide negative 3 by 5 to convert this fraction to a decimal
- The x-intercept should then equal -0.6
Converting Slope Intercept Form To Standard Form
In an equation that relates the first power of x to the first power of y, there will be a straight line on an x-y graph. To express that as a standard form, it would look like Ax + By + C = 0, or Ax + By = C. When this equation is rearranged with y by itself on the left side, it would look like y = mx + b.
This is the slope intercept form, with m being equal to the slope of the line and b being the value of y when x =0, making it the y-intercept. Rearranging an x intercept to standard form, then, isn’t much more than just simple arithmetic.
- Subtract mx from both sides: y – mx = (mx – mx) +b, so y = mx = b
- Subtract b from both sides: y – mx – b = b – b, so y – mx – b = 0
- Rearrange with x term first: -mx + y – b = 0
- If the fraction A/B represents m, the equation should be -A/Bx + y – b = 0
- Multiply both sides of the equation using the denominator B: -Ax +By – Bb = 0
- If Bb = C, the equation should be -Ax + By – C = 0
Using a graphing calculator to address a quadratic equation or simply as an x intercept calculator means having the ability to generate a graph of y =f(x) and the capacity to scale the graph for the size of the graphing surface. It also means being able to know that when f(x) = 0, x is the solution to the equation.
Graphing calculators may require the user to positioning a cursor at an approximate value for the root, and others might require brackets for the root on either side of the zero. The calculator then uses its algorithms to compute the proper position of the root, to the limits of its own accuracy.