Question

Use a graphing calculator to approximate the solution of the equation. $$ x^{2}-3 x-4=0 $$

Short answer

The solutions of the quadratic equation \(x^{2}-3x-4=0\) are approximately \(x=4\) and \(x=-1\), obtained using a graphing calculator.

Step-by-step solution

Understand the Problem

Consider the quadratic equation \(x^{2}-3x-4=0\). This will graph a parabola and the zero(s) of the quadratic -- i.e., x for which \(f(x)=0\)-- are the x-values at which the parabola intersects the x-axis.

Input Equation Into Calculator

Input the equation \(x^{2}-3x-4\) into the graphing calculator's function input field. Make sure the graphing mode is set correctly (usually 'function' mode for such equations).

Plot the Graph

Use the plot function to graph the quadratic equation. The graph should be a roughly 'U' or inverted 'U' shaped curve, called a parabola.

Identify the Roots

View the points where the graph intersects the x-axis. These points are the roots of the equation -- i.e., when \(y=0\). Use trace or intersect or zeros finder function (depending upon the calculator model) to get approximate values of x at the points where the graph touches the x-axis.

Key concepts

Graphing Calculator

A graphing calculator is an advanced digital tool that allows students to visualize and solve various mathematical problems, especially those involving equations and functions. To use a graphing calculator effectively for solving quadratic equations, one initially inputs the equation into the designated function field. After entering the quadratic expression, such as x^2 - 3x - 4, the calculator can graph the equation revealing its characteristic shape on a coordinate plane. Graphing calculators usually come equipped with features like trace, zoom, and intercepts to help pinpoint the exact coordinates of important features on the graph, such as the roots or vertex of a parabola.

Utilizing these functions in a graphing calculator can simplify the process of solving complex equations by transforming them into visual problems. Working with a visual representation often makes it easier for students to understand the relationships between algebraic expressions and their graphical outputs.

Parabola

The graph of any quadratic equation like x^2 - 3x - 4 = 0 is a curve known as a parabola. This shape is symmetrical and can open upwards or downwards depending on the sign of the coefficient of the squared term in the equation. For our example, since the coefficient of x^2 is positive, the parabola opens upwards, resembling a 'U'.

Properties of a Parabola

Parabolas have a vertex, which is the highest or lowest point on the graph, and an axis of symmetry, which is a vertical line that divides the parabola into two mirror-image halves. The roots of the quadratic equation correspond to the points where the parabola intersects the x-axis, also known as the x-intercepts. Understanding the shape and properties of a parabola is crucial because it provides a visual context for the solutions of the quadratic equation.

Roots of Quadratic

The roots of a quadratic equation are the values of x that satisfy the equation, meaning they make the equation true when substituted in place of x. In graph terms, these are the x-values where the parabola crosses the x-axis. There can be two, one, or no real roots, depending on whether the parabola touches the x-axis twice, once (vertex touching the x-axis), or not at all, respectively.

Finding the Roots

To find the roots using a graphing calculator, you observe where the graph intersects the x-axis. The coordinates of these intersection points have a y-value of 0, and their x-values are the roots we are interested in. For the equation x^2 - 3x - 4 = 0, there will be two roots since it is a simple quadratic equation with a positive leading coefficient and a negative constant term.

Equation Intersection with X-axis

When discussing the equation intersection with the x-axis, we are referring to the points at which the graph of the equation crosses the horizontal axis of the coordinate system. These intersection points are pivotal as they represent the solutions to the equation when set equal to zero, which is the fundamental basis for finding roots.

Intersecting the X-axis

In the context of a quadratic equation, these points are where the output value (y) of the function is zero. By analyzing these intersection points on a graph, one can determine not only the number of real roots but also their approximate values. Graphing calculators make this process much more efficient by providing features for calculating, and sometimes even graphically marking, these crucial intersection points.