Problem 25 - Algebra and Trigonometry

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$ y=9-x^{2} $$

Step-by-Step Solution

Answer

After substituting the value of x into the equation \(y = 9 - x^2\), for each of the given x-values, plot the points and connect them. The graph obtained is a downward facing parabola.

Step 1: Substitute the given x-values into the equation to find the corresponding y-values

For \(x = -3, -2, -1, 0, 1, 2, 3\), substitute each of these values into the equation \(y = 9 - x^2\), to get: for \(x = -3, y = 0\), for \(x = -2, y = 5\), for \(x = -1, y = 8\), for \(x = 0, y = 9\), for \(x = 1, y = 8\), for \(x = 2, y = 5\), and finally for \(x = 3, y = 0\).

Step 2: Plot the calculated points on the graph

The next step is to mark these calculated points (-3,0), (-2,5), (-1,8), (0,9), (1,8), (2,5), (3,0) on the graph. Connect these points with a smooth curve.

Step 3: Interpret the graph

The curve obtained is a downward facing parabola. This is due to the negative sign attached to the square term in the original equation. This signifies that as x increases or decreases from 0, y decreases.

Key Concepts

Graphing Quadratic Functions

In the realm of mathematics, graphing quadratic functions is a fundamental skill in understanding the behavior of equations of the form \[ y = ax^2 + bx + c \].

For the exercise, the function given was \[ y = 9 - x^2 \].This is a specific form of a quadratic known as the vertex form, where the vertex is at the point (0,9). Graphing these functions involves determining how the parabola - the U-shaped graph - will open and where it is located on the coordinate plane.

For the equation \[ y = 9 - x^2 \], our focus is centered on understanding that the highest point of the parabola (its vertex) is where \[ x = 0 \]. Beyond this point, the values of \( y \) will decrease.

Plotting Points

Plotting points is the initial step needed for drawing an accurate graph of a quadratic equation. By selecting specific \( x \)-values such as \[ -3, -2, -1, 0, 1, 2, \text{and} 3 \], we can plug these numbers into the function \[ y = 9 - x^2 \]to calculate the corresponding \( y \)-values. This simple substitution results in coordinates:

• For \( x = -3 \), \( y = 0 \).
• For \( x = -2 \), \( y = 5 \).
• For \( x = -1 \), \( y = 8 \).
• For \( x = 0 \), \( y = 9 \).
• For \( x = 1 \), \( y = 8 \).
• For \( x = 2 \), \( y = 5 \).
• For \( x = 3 \), \( y = 0 \).

These points will provide a guide for creating the parabolic shape on the graph.

Interpreting Graphs

Interpreting graphs involves analyzing the plotted curve to extract valuable insights about the quadratic function. For the graph created by \[ y = 9 - x^2 \], the key observation is the symmetric nature of the plotted parabola. With the vertex at the highest point \( (0,9) \), the graph reveals symmetry about the vertical line \( x = 0 \).

This symmetry is a defining characteristic of all parabolas derived from a quadratic equation. The graph also shows how function values decrease as \( x \) moves away from the vertex in either direction - a feature caused by the negative coefficient of the \( x^2 \) term. It's important to grasp the notion that this dictates the direction in which the parabola opens.

Quadratic Equations

Quadratic equations are a type of polynomial equation of the form \[ ax^2 + bx + c = 0 \].In the context of the exercise, the equation \[ y = 9 - x^2 \]is a simple example where only the \( ax^2 \) and \( c \) terms are present.

This showcases a standard type of quadratic equation, where the graph portrays a parabola. Depending on the coefficient \( a \) of the \( x^2 \) term, the parabola will either open upwards (when \( a > 0 \)) or downwards (when \( a < 0 \)). In this exercise, due to \( a = -1 \), the parabola opens downwards. By solving these equations graphically or algebraically, one can find the roots or solutions, often connected to where the parabola intersects the \( x \)-axis.