Problem 35 - Algebra 2

Question

LAWN CARE For Exercises 33 and \(34,\) use the following information. The path of water from a sprinkler can be modeled by a quadratic function. The three functions below model paths for three different angles of the water. $$\begin{array}{l}{\text { Angle } \mathrm{A} : y=-0.28(x-3.09)^{2}+3.27} \\\ {\text { Angle } \mathrm{B} : y=-0.14(x-3.57)^{2}+2.39} \\ {\text { Angle } \mathrm{C} : y=-0.09(x-3.22)^{2}+1.53}\end{array}$$ Which sprinkler angle will produce the widest path? The narrowest path?

Step-by-Step Solution

Answer

Angle C produces the widest path; Angle A produces the narrowest path.

Step 1: Identify the Standard Form of a Parabola

Each quadratic function provided represents a parabola. The standard form here is \( y = a(x-h)^2 + k \), where \( a \) determines the shape, and \( (h,k) \) is the vertex.

Step 2: Analyze the Coefficient 'a' for Each Function

The coefficient \( a \) dictates the width of the parabola: smaller absolute values of \( a \) result in wider paths, and larger absolute values of \( a \) result in narrower paths. Identify \( a \) for each function: Angle A: \( a = -0.28 \), Angle B: \( a = -0.14 \), Angle C: \( a = -0.09 \).

Step 3: Compare the Absolute Values of 'a'

Compare the absolute values of \( a \) for all three angles to determine the width of each path: \(|-0.28| = 0.28\), \(|-0.14| = 0.14\), and \(|-0.09| = 0.09\). The smallest absolute \( a \) value represents the widest path.

Step 4: Determine the Widest Path

The smallest absolute value of \( a \) is \( 0.09 \), corresponding to Angle C. Therefore, Angle C will produce the widest path.

Step 5: Determine the Narrowest Path

The largest absolute value of \( a \) is \( 0.28 \), corresponding to Angle A. Therefore, Angle A will produce the narrowest path.

Key Concepts

Parabola

A parabola is a beautiful U-shaped curve that you often see in the graphs of quadratic functions. Imagine the water from the sprinkler arching beautifully and landing to form this curved shape that we call a parabola. This happens because quadratic functions always plot into this specific shape. Parabolas are symmetric and have important properties that determine their width and direction. The way a parabola opens depends on its equation’s specific features. Some parabolas can open upwards, like a smile, or downwards, like a frown. In these exercises, we're looking at downward-opening parabolas, representing the path of the water from a sprinkler.

Coefficient

In a quadratic equation, the coefficient can tell us a lot! Specifically, we're exploring the coefficient known as 'a' in the equation form, which looks like this:

• Standard form: \( y = a(x-h)^2 + k \)

The coefficient 'a' controls the parabola's width and direction. A larger absolute value of 'a' means the parabola will be "narrower," whereas a smaller absolute value will make it "wider." Think about squishing or stretching a balloon vertically. If you press down hard in the middle, you get a narrow shape; if you lightly tap, it widens. For the sprinkler problem, checking 'a' helps to see how much the water will spray outwards or focus closely.

Vertex

The vertex of a parabola is like its heart. It is the peak or the lowest point of the curve, depending on its direction. For a downward-opening parabola, the vertex is the highest point. The position of the vertex is given by the coordinates \( (h,k) \) in our standard form equation:

• \( y = a(x-h)^2 + k \)

This point is where the path of the water is at its maximum height before it descends back to the ground. In our sprinkler paths, these vertex points tell us where the water peaks. Each angle's vertex may change depending on the values of 'h' and 'k,' shifting the curve's height and position along the x-axis.

Quadratic equation

A quadratic equation is a type of polynomial equation where the highest exponent of a variable is a square. It typically appears in the form

• \( ax^2 + bx + c = 0 \)

This form shows how different parts of the equation influence its graph, leading to the parabolas that we've been examining.
Quadratic equations can have several real-world applications, such as predicting the path of a thrown ball, or in our case, the arc of water from a sprinkler. By manipulating values of 'a', 'b', and 'c' or converting into the vertex form \( y = a(x-h)^2 + k \), we can better understand the behavior of the equation and its graphical representation. In these problems, the quadratic form helps us visualize how different angles impact the water spray path.