Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let z-represent a positive real number. Describe how the family of parabolas represented by the given function compares with the graph of \(g(x)=x^{2}\) \(f(x)=z x^{2}+4\)

Step-by-Step Solution

Answer

The family of parabolas represented by \(f(x) = zx^{2} + 4\) can be broader, narrower or the same as \(g(x) = x^{2}\) depending on the value of z. Additionally, the graph of \(f(x)\) is vertically shifted up by 4 units due to the additional constant term.

Step 1: Analyze the Effect of 'z'

First, consider the implication of factor z in the function \(f(x) = zx^{2} + 4\). The factor z affects the 'width' of the parabola. When z > 1, the parabola narrows and when 0 < z < 1, the parabola widens. On the other hand, when z = 1 the parabola remains the same as the graph of \(f(x) = x^{2}\).

Step 2: Analyze the Effect of 4

Now, examine the addition of a constant term, 4, in \(f(x) = zx^{2} + 4\). The constant 4 results in a vertical shift of the graph of the parabola by 4 units. This means the Vertex of the new parabola \(f(x)\) is at point (0,4) compared to the parabola of \(g(x)\) whose vertex is at point (0,0).

Step 3: Combine Effects

Finally, with above mentioned transformation both changes occur simultaneously in \(f(x) = zx^{2} + 4\). Namely, the parabola 'width' changes based on the parameter z and the entire graph moves 4 units up due to the added constant

Key Concepts

Real Numbers

In mathematics, real numbers include all numbers on the number line. This encompasses both rational numbers, like integers and fractions, and irrational numbers, such as \( \pi \) and \( \sqrt{2} \). Real numbers are essential in algebra as they allow for a wide range of mathematical operations.
They are crucial when studying functions and transformations, as they can represent coefficients, constants, and variables in equations.

• **Positive Real Numbers**: These are numbers greater than zero, which play a particular role as they are often coefficients, like 'z' in a given quadratic function, affecting its properties.
• **Properties**: Real numbers are ordered, meaning any two different numbers can be compared, and they have no gaps on the number line.

Understanding real numbers enables the study of continuous functions and allows mathematicians to apply concepts like limits and derivatives.

Vertical Shift

In function transformations, a vertical shift changes the graph of a function upwards or downwards in relation to its original position. This occurs when a constant is added or subtracted from the function. For the quadratic function \( f(x) = zx^2 + 4 \), the addition of 4 indicates a vertical shift.
This means the entire graph moves 4 units up.

• **Vertex Change**: The vertex of the original parabola \( g(x) = x^2 \) is at (0,0). After the shift, the vertex for \( f(x) \) will be at (0,4).
• **Graph Behavior**: This shift doesn't alter the shape but impacts the location of the minimum or maximum point on the graph.

A vertical shift is significant in that it affects the output of the function without impacting the 'steepness' or 'width' of a parabola.

Function Transformation

Function transformation modifies a graph in specific ways: altering its position or shape. These transformations include translations, reflections, stretches, and compressions. For the function \( f(x) = zx^2 + 4 \), both vertical shifts and changing the factor 'z' are key transformations.

• **Translation**: This shifts the entire graph, in this case, vertically, due to the constant addition.
• **Stretch and Compression**: Altering 'z' changes the degree to which the parabola stretches or compresses. A larger 'z' narrows the parabola, while a smaller one widens it.

Understanding these transformations enable the construction of complex functional graphs. They show how simple changes in a function's equation translate into shifts or alterations in their graphical representations.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, usually presented in the form \( ax^2 + bx + c \). The basic form \( g(x) = x^2 \) is a simple parabola opening upwards.
In \( f(x) = zx^2 + 4 \), the parabola's equation is altered by changing 'a' to 'z', and adding a constant shifts it vertically.

• **Key Characteristics**: Parabolas always have a symmetric shape, a vertex, and they can open upward or downward depending on the leading coefficient's sign.
• **Applications**: These functions appear in various real-world scenarios, from projectile motion calculations to financial models.

Mastery of quadratic functions involves understanding how their general form can be adjusted through specific coefficients and constants, resulting in different graphical behaviors.