Use a graphing calculator to solve each system. Then confirm your answer algebraically. y=2

Use a graphing calculator to solve each system. Then confirm...

Key Concepts

Algebraic Verification

Algebraic verification involves using analytical methods, such as substitution or elimination, to solve equations and confirm the solutions obtained by graphical methods. This process is essential as it not only verifies the accuracy of the graphical solutions but also provides exact numerical answers and insights into the behavior of the functions involved.

Intersection of Graphs

The intersection of graphs of two functions represents the set of points at which both function values are equal. For systems involving quadratic functions, finding these points graphically can provide an intuitive understanding of the number and approximate location of solutions, which can then be confirmed through algebraic methods.

Quadratic Functions

Quadratic functions are second-degree polynomial functions that produce parabolic graphs. Their general form is expressed as y = ax^2 + bx + c. Understanding the properties of quadratic functions, such as their symmetry, vertex, and direction of opening, is fundamental when analyzing the solutions of systems that include quadratic equations.

Graphing Calculators

Graphing calculators are powerful tools that allow users to visually represent and analyze functions and equations. They can be used to plot curves, zoom in and out to inspect details, and identify approximate points of intersection, which are crucial when solving systems of equations that involve nonlinear functions.

Systems of Equations

A system of equations is a collection of two or more equations with a common set of variables. The solution to the system is the set of points that simultaneously satisfy all equations. In the context of quadratic and other nonlinear functions, these systems can have multiple, single, or no solutions depending on how the graphs intersect.

Transcript

00:01 Okay, here's our system of equations and we're going to start by using a graphing calculator to solve the system and then come back and answer algebraically and make sure that those answers agree.

00:11 So here's my graphing calculator and i went into the y equals menu and there i typed y1 and y2 the two equations and then i'm using a window that shows me from roughly negative 6 to 6 and negative 4 to 4 on my x and y axes and here we see the two parabolas they do intercept we have two intersection points.

00:32 Let's go ahead and find the coordinates of those points.

00:35 So i'm going to go into the calculate menu.

00:37 Choose number 5, intersect.

00:39 First curve yes, second curve yes, and then we'll give it a guess.

00:44 So the intersection point closest to the right looks like it's at about negative 1 3rd, negative 1 in 1 9th, i'm going to guess.

00:55 And then for the other one, back into the calculate menu, choose intersect again.

00:59 First curve yes, second curve yes.

01:03 Then we move the cursor over closer to the other point.

01:06 And we get negative 1, negative 2.

01:09 So let's confirm those algebraically.

01:12 We can use substitution and substitute the value for y from the second equation into the first equation.

01:19 We have the opposite of x squared minus 1 equals 2x squared plus 4x.