Problem 73 - Algebra and Trigonometry

Question

A family of functions is given. In parts (a) and (b) graph all the given members of the family in the viewing rectangle indicated. In part (c) state the conclusions you can make from your graphs. $$f(x)=x^{2}+c$$ (a) \(c=0,2,4,6 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-2,-4,-6 ;[-5,5]\) by \([-10,10]\) (c) How does the value of \(c\) affect the graph?

Step-by-Step Solution

Answer

The value of \(c\) shifts the parabola vertically by \(c\) units. Positive \(c\) shifts up, negative \(c\) shifts down.

Step 1: Understanding the Function Family

The function given is \( f(x) = x^2 + c \). This is a parabola that opens upwards with a vertex located at the point \((0, c)\). The parameter \(c\) represents the vertical shift of the standard parabola \( y = x^2 \).

Step 2: Part A: Graphing for c = 0, 2, 4, 6

For each value of \(c\), the graph is a parabola: - For \(c=0\), the parabola is \(y = x^2 \), with vertex at \( (0,0) \). - For \(c=2\), the parabola is \(y = x^2 + 2 \), shifted up by 2 units, vertex at \((0,2)\). - For \(c=4\), the parabola is \(y = x^2 + 4 \), shifted up by 4 units, vertex at \((0,4)\). - For \(c=6\), the parabola is \(y = x^2 + 6 \), shifted up by 6 units, vertex at \((0,6)\). Graph these within the viewing rectangle \([-5,5]\) by \([-10,10]\).

Step 3: Part B: Graphing for c = 0, -2, -4, -6

For each value of \(c\), the graph remains a parabola: - For \(c=0\), the parabola is \(y = x^2 \), as in part (a), with vertex at \((0,0)\).- For \(c=-2\), the parabola is \(y = x^2 - 2 \), shifted down by 2 units, vertex at \((0,-2)\).- For \(c=-4\), the parabola is \(y = x^2 - 4 \), shifted down by 4 units, vertex at \((0,-4)\).- For \(c=-6\), the parabola is \(y = x^2 - 6 \), shifted down by 6 units, vertex at \((0,-6)\).Graph these within the viewing rectangle \([-5,5]\) by \([-10,10]\).

Step 4: Part C: Conclusion on the Effect of c

The parameter \( c \) represents a vertical shift of the parabola \(y = x^2\). As \(c\) increases, the entire graph shifts upward by \(c\) units; as \(c\) decreases, it shifts downward by \(c\) units. The shape of the parabola or the direction it opens does not change with \(c\).

Key Concepts

Parabola

A parabola is a special type of curve on a graph. It's familiar because of its distinctive U-shape. The basic equation that defines a parabola is \(y = x^2\). This is the simplest form of a parabola, and it opens upward. Parabolas are part of a bigger group called quadratic functions.
They can open either upward or downward depending on their equations. However, in the case of \(f(x) = x^2 + c\), all parabolas open upward because the term containing \(x^2\) has a positive coefficient.
This shape is symmetrical around a vertical line, called the axis of symmetry. For the function \(f(x) = x^2 + c\), this line is the \(y\)-axis, or \(x = 0\). All parabola graphs of this form will visually look the same, just shifted up or down.
Understanding parabolas is crucial because they model various real-world phenomena, such as the path of a basketball or the design of satellite dishes.

Vertex

In any quadratic function, the vertex is a special point where the parabola reaches its highest or lowest point. For the equation \(f(x) = x^2 + c\), the vertex is at \((0, c)\). Here, \(0\) comes from the \(x\)-coordinate, indicating the axis of symmetry, and \(c\) is the \(y\)-coordinate, pinpointing the height or depth of the vertex.
The vertex is significant: it tells us the minimum value of the parabola when it opens upward, or the maximum value if it opens downward. For equations like \(f(x) = x^2 + c\), where the parabola opens upwards, the vertex represents the lowest point on the graph.
By observing the vertex, you immediately know the turning point's location. Since all graphs of the form \(x^2 + c\) share the same symmetry and basic shape, only their positions vary depending on \(c\).
This understanding helps in graphing functions quickly and provides insights into how altering the equation affects the graph.

Vertical Shift

The vertical shift refers to how a graph of a function moves up or down on a graphing plane. In the equation \(f(x) = x^2 + c\), \(c\) is responsible for this vertical movement.
When \(c\) is positive, the whole parabola moves up by \(c\) units; this means the graph is lifted higher on the \(y\)-axis but keeps its shape and opening direction. Conversely, if \(c\) is negative, the parabola shifts down by \(c\) units.
The actual curve doesn't change. It remains the familiar U-shape, maintaining the same width and direction. The key to understanding the vertical shift is to remember that it doesn't affect how "wide" or "narrow" the curve is, only where it sits on the graph.
This concept is handy because it simplifies graph transformations: without changing the essence of the curve, you can represent a wide range of functions by simply adjusting \(c\).

Graphing Functions

Graphing functions is a method to visually represent mathematical relationships. It offers a picture of how values are related, especially for equations like \(f(x) = x^2 + c\).
To graph these functions, start by plotting the vertex, \((0, c)\). Then sketch the parabola emerging from this point, symmetrically curving upwards. Adjust the graph according to the value of \(c\): positive values lift the graph up, and negative values bring it down.
The boundaries for plotting are given as \([-5, 5]\) for \(x\) and \([-10, 10]\) for \(y\). This tells you where to begin and end your graph on the plane. Creating these visual representations helps in understanding key features like symmetry, maximum, minimum points, and the influence of parameters like \(c\). Graphing is a fundamental tool in math for analyzing and predicting changes in data, providing clarity beyond just numbers.