Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given \(x\) -intercepts. (There are many correct answers.) (0, 0), (10, 0)

Step-by-Step Solution

Answer

The quadratic function that opens upwards with roots (0,0) and (10,0) is \( y=x^2-10x \) and the function that opens downwards is \( y=-x^2+10x \).

Step 1: Understanding the Roots

When a polynomial is expressed in the form \( y=a(x-r1)(x-r2) \), \( r1 \) and \( r2 \) are the roots or x-intercepts. In our case, we are given that the roots are 0 and 10.

Step 2: Forming the Quadratic Function that Opens Upwards

Let's take \( a=1 \) for our first function. Substituting \( r1=0 \), \( r2=10 \) and \( a = 1 \) in \( a(x-r1)(x-r2) \), we get \( y1=1*(x-0)*(x-10) = x(x-10) = x^2-10x \).

Step 3: Forming the Quadratic Function that Opens Downwards

Next, let's take \( a=-1 \) for our second function. Substituting \( r1=0 \), \( r2=10 \) and \( a = -1 \) in \( a(x-r1)(r2-x) \), we get \( y2=-1*(x-0)*(x-10) = -x(x-10) = -x^2+10x \).

Key Concepts

X-Intercepts of Quadratics

Understanding the x-intercepts of quadratic functions is foundational for graphing these curves and analyzing their properties. The x-intercepts, also known as zeros or roots, are the points where the graph of the quadratic equation crosses the x-axis. These intercepts are the solutions to the equation when we set the output value, i.e., the y-coordinate, to zero.

For example, with the quadratic function given in our exercise, the x-intercepts are at (0, 0) and (10, 0). This means if you plug in x = 0 or x = 10 into the quadratic equations, the output will be zero. Finding the x-intercepts is an essential step in sketching the graph and is also used in many application problems involving quadratics.

It's important to note that a quadratic can have either two, one, or no real x-intercepts depending on whether the parabola crosses the x-axis twice, touches it once, or does not touch it at all. This is determined by the discriminant in the quadratic equation.

Opening Direction of Parabolas

The direction in which a parabola opens is pivotal in understanding the behavior of quadratic functions. Parabolas are the graphs formed by quadratic equations, and they can either open upwards, like a cup, or downwards, like an upside-down cup.

The sign of the coefficient 'a' in the quadratic equation in standard form, which is written as \( y = ax^2 + bx + c \), determines this opening direction. When 'a' is positive, the parabola opens upward. Conversely, when 'a' is negative, the parabola opens downward. In our exercise, for instance, when we have \( a=1 \), the corresponding parabola would open upwards, forming a minimum point. When \( a=-1 \), the parabola opens downwards, forming a maximum point.

This characteristic is crucial when it comes to the maximum or minimum values that can be obtained from a quadratic function, which have practical implications in optimization problems across various disciplines.

Factored Form of Quadratic Equations

The factored form of a quadratic equation provides a direct way to identify the x-intercepts of the function. This form is particularly useful because it is easier to solve and visualize the roots of the equation.

The factored form is expressed as \( y = a(x-r_1)(x-r_2) \), where 'a' is a non-zero coefficient, and \( r_1 \) and \( r_2 \) are the x-intercepts of the parabola. In our exercise, we transformed the given x-intercepts, 0 and 10, into the factored form of two different quadratic functions, one opening upward and one downward.

By using the factored form, we are able to quickly determine the structure of the quadratic function and graph it accordingly. Additionally, the factored form also emphasizes the symmetric nature of parabolas, as the axis of symmetry lies exactly halfway between the two x-intercepts.