Problem 56 - A Graphical Approach to College Algebra

Question

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a parabola; two points.

Step-by-Step Solution

Answer

Choose \(y = x^2\) for the parabola and \(y = 2x + 1\) for the line to ensure two intersections.

Step 1: Understanding the Problem

We need to draw a sketch of a line and a parabola such that they intersect at exactly two distinct points.

Step 2: Identifying Functions

Let's consider a standard parabola, such as \(y = x^2\), and a line, such as \(y = mx + c\). Our task is to adjust the line so it intersects the parabola at two points.

Step 3: Setting Up the Equation

To find the intersection points, set the functions equal: \(x^2 = mx + c\). This results in a quadratic equation \(x^2 - mx - c = 0\).

Step 4: Determining the Number of Intersections

A quadratic equation has two distinct real roots if its discriminant \(b^2 - 4ac\) is positive. Here, \(a = 1\), \(b = -m\), and \(c = -c\), so the discriminant is \((-m)^2 - 4 \, (1) \, (-c) = m^2 + 4c\). Ensure that \(m^2 + 4c > 0\).

Step 5: Adjusting Parameters

Choose parameters for the line. Suppose \(m = 2\) and \(c = -3\). Check the discriminant \(m^2 + 4c = 2^2 + 4(-3) = 4 - 12 = -8\), which is negative. Adjust to \(m = 2\) and \(c = 1\), for \(m^2 + 4c = 4 + 4 = 8 > 0\).

Step 6: Sketching the Graphs

Plot the parabola \(y = x^2\) and the line \(y = 2x + 1\). Ensure the line crosses the parabola at two points, confirming they intersect at two distinct points.

Key Concepts

Intersection Points

Intersection points are key places where two graphs, like a line and a parabola, meet. In geometry, finding intersection points helps us understand how different functions relate to each other visually.
To visualize this, imagine a simple parabola represented by the equation \(y = x^2\) and a line such as \(y = mx + c\). The task is to adjust these functions so that they intersect at two distinct points. These points of intersection indicate the solutions of the equations set equal to each other. In this scenario: \(x^2 = mx + c\).
When sketching, we aim to bring the line across the parabola, ensuring the graphs meet at exact positions, verifying our understanding of how these mathematical relationships unfold visually.

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. It has the standard form: \(ax^2 + bx + c = 0\). In our scenario, setting the equation of the line equal to the parabola \(x^2 = mx + c\) results in the quadratic equation \(x^2 - mx - c = 0\).
This equation is crucial because it gives us the algebraic pathway to understand where and how these graphs intersect. By solving the quadratic equation, we identify the values of \(x\) where these intersection points lie on the graph. These solutions, therefore, represent the x-coordinates of the intersection points.
Mastering quadratic equations empowers you to decode various geometric intersections, not just lines and parabolas.

Discriminant

The discriminant is a special part of the quadratic formula \(b^2 - 4ac\), derived from the quadratic equation \(ax^2 + bx + c = 0\). It determines the nature of the solutions to a quadratic equation.
In our case of finding intersection points, the discriminant of \(x^2 - mx - c = 0\) becomes \((-m)^2 - 4 \times 1 \times (-c) = m^2 + 4c\). This value reveals how many real roots, or solutions, the equation has. A positive discriminant means we expect two distinct real roots, which correlate with the two intersection points of the line and the parabola.
Understanding the discriminant helps predict the behavior of the graph even before solving the equation entirely.

Real Roots

Real roots are the solutions to a quadratic equation that are real numbers. For a graph, these roots correspond to the x-values where the intersection occurs. They help us determine where exactly along the x-axis the line and parabola share points.
In our exercise, for the equation \(x^2 - mx - c = 0\), the condition for real roots is that the discriminant \(m^2 + 4c\) must be greater than zero. This ensures that the intersection points are true and verifiable on the graph.
Real roots confirm that the mathematical setup of our functions displays intersections that are grounded in real, observable coordinates on the plotted graphs. Not having real roots would indicate no intersection on the graph in our real-numbered plane.