Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Write an inequality that satisfies the description.Below the parabola with vertex \((0,1)\) and \(x\) -intercepts \((-1,0)\) and \((1,0)\).

Step-by-Step Solution

Answer

The inequality is \( y < -x^2 + 1 \).

Step 1: Understand the Parabola

The parabola has a vertex at (0,1) and x-intercepts at (-1,0) and (1,0). Thus, the equation of the parabola is of the form \( y = a(x+1)(x-1) \) since it is symmetrical about the y-axis.

Step 2: Find the Equation of the Parabola

Because the vertex is at (0,1), we set \( x = 0 \) in the equation \( y = a(x^2 - 1) = 1 \). This simplifies to \( y = a(0^2 - 1) = 1 \Rightarrow -a = 1 \). Thus, \( a = -1 \). The equation for the parabola is \( y = -(x^2 - 1) = -x^2 + 1 \).

Step 3: Write the Inequality

We need to find the inequality that represents the region below this parabola. Below the parabola means \( y \) values are less than the quadratic function. Thus, the inequality is \( y < -x^2 + 1 \).

Key Concepts

Vertex Form

The vertex form of a quadratic function provides an easy way to identify the vertex of a parabola. The vertex form is written as \( y = a(x-h)^2 + k \), where \((h,k)\) are the coordinates of the vertex. By using this form, one can quickly determine the peak or the lowest point on the graph of a parabola. In this specific case, the vertex of the parabola is given as \((0,1)\). This means \(h = 0\) and \(k = 1\). This form is particularly useful for visualizing and graphing parabolas, as it clearly shows where the vertex lies on the coordinate plane. If you know the vertex of a parabola, you can derive its equation efficiently and understand how it is positioned relative to the x-axis and y-axis.

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis. For quadratic functions, these are also known as the roots or zeroes of the function. In our exercise, the parabola has x-intercepts at \((-1,0)\) and \((1,0)\). This means that when \(x = -1\) and \(x = 1\), the value of \(y\) is zero. These intercepts are vital in writing quadratic equations because they indicate the points where the parabola touches or crosses the x-axis. To construct a quadratic equation given the x-intercepts, you can use the factored form \( y = a(x-x_1)(x-x_2) \), where \(x_1\) and \(x_2\) are the x-intercepts. Knowing these intercepts helps us build the equation and understand how the parabola behaves on the x-y plane.

Quadratic Function

Quadratic functions take the form \( y = ax^2 + bx + c \). They describe parabolas, which can open upward or downward depending on the sign of \(a\). In our case, the quadratic equation derived is \( y = -x^2 + 1 \). Here, \(a = -1\), which indicates that the parabola opens downward. The term \(x^2\) describes the quadratic behavior, while the constants \(b\) and \(c\) determine the parabola's shift along the axis. Quadratic functions are fundamental in mathematics because they appear in various real-world contexts, such as physics and economics. Understanding their structure allows one to predict and analyze the motion and relationships described by the function.

Inequality Solutions

An inequality involving a quadratic function defines a range of values rather than an explicit equation. When tasked with finding the inequality for a region below a parabola, we seek values where \(y\) is less than the quadratic function. In this problem, the inequality \( y < -x^2 + 1 \) represents all the points where the value of \(y\) is beneath the curve of the parabola \( y = -x^2 + 1 \). Solving quadratic inequalities helps in understanding and visualizing regions on the coordinate plane. These inequalities are useful in various practical scenarios, such as determining potential outcomes in economic models or modeling structures in engineering.