Problem 70 - 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 that you can make from your graphs. \(f(x)=(x-c)^{2}\) (a) \(c=0,1,2,3 ; \quad[-5,5]\) by \([-10,10]\) (b) \(c=0,-1,-2,-3 ; \quad[-5,5]\) 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 horizontally; to the right if \( c \) is positive, and to the left if negative.

Step 1: Understanding the Function

The family of functions given is in the form \( f(x) = (x - c)^2 \). This is a quadratic function, and its graph is a parabola opening upwards. The parameter \( c \) shifts the graph horizontally.

Step 2: Graphing Part (a) – Positive Shifts

For part (a), where \( c = 0, 1, 2, 3 \), plot each function: \( f(x) = x^2 \), \( f(x) = (x - 1)^2 \), \( f(x) = (x - 2)^2 \), and \( f(x) = (x - 3)^2 \) in the given viewing rectangle \([-5,5]\) by \([-10,10]\). Each graph is a parabola that shifts right with increasing \( c \).

Step 3: Graphing Part (b) – Negative Shifts

For part (b), where \( c = 0, -1, -2, -3 \), plot each function: \( f(x) = x^2 \), \( f(x) = (x + 1)^2 \), \( f(x) = (x + 2)^2 \), and \( f(x) = (x + 3)^2 \) in the same viewing rectangle. In this case, each graph is a parabola that shifts left with decreasing \( c \).

Step 4: Analyzing Vertical Position and Shape

Observe that regardless of \( c \), all the parabolas open upwards and share the same shape (width), they only differ in horizontal position.

Step 5: Conclusion on the Effect of \( c \)

The parameter \( c \) affects the horizontal position of the parabola's vertex. Increasing \( c \) shifts the parabola to the right, while decreasing \( c \) shifts it to the left. The vertex aligns at \( (c, 0) \) on the \( x \)-axis.

Key Concepts

Parabola

In the world of quadratic functions, the graph of such functions usually takes the form of a curve called a **parabola**. A parabola is a symmetrical curve that can open upwards or downwards. In the case of the function \( f(x) = (x-c)^2 \), the parabola opens upwards, forming a "U" shape. Parabolas have some interesting properties: - They have an axis of symmetry, which is a vertical line that goes through the middle of the parabola. For our function, the axis of symmetry is \( x = c \). This means each side of the parabola is a mirror image of the other. - The parabola is continuous and extends infinitely in both directions, up and down. - The width of the parabola is determined by the coefficient of \( x^2 \). Here, since it is 1, all parabolas are consistently shaped and only shift horizontally without altering their size or orientation. Understanding parabolas is essential for mastering quadratic functions. With practice, you'll become familiar with how changes in the function's equation modify the graph's shape and position.

Vertex

The **vertex** is a crucial feature of a parabola and is essentially the turning point of the curve. For a parabola like \( f(x) = (x-c)^2 \), the vertex is located at the point \( (c, 0) \). This point is where the parabola reaches its minimum value, as it opens upwards. Key points about the vertex: - It is the lowest point on the parabola when it opens upwards and the highest when it opens downwards (though this doesn't apply in our current scenario). - The vertex lies on the axis of symmetry, making it the central point of the parabola. - In the function \( f(x) = (x-c)^2 \), the vertex being at \( (c, 0) \) tells us that the function's smallest output is at \( x = c \), showing the importance of locating the vertex to understand the parabola's behavior fully. When you graph quadratic functions, the vertex provides a reference for plotting the curve accurately and understanding how changes to the function affect its shape.

Horizontal Shift

One of the fascinating transformations affecting quadratic functions like \( f(x) = (x-c)^2 \) is the **horizontal shift**. This type of transformation moves the entire parabola left or right on the coordinate plane. Important details about horizontal shifts: - The horizontal shift is dictated by the value of \( c \) in \( (x-c)^2 \). - If \( c \) is positive, the parabola shifts to the right by \( c \) units. Conversely, if \( c \) is negative, the parabola moves to the left by \(|c|\) units. This is because the expression \( (x-c) \) means setting the vertex at \( x = c \), reflecting the shift direction. - Regardless of how \( c \) changes, the shape and orientation of the parabola remain unaffected; it simply relocates its position.Comprehending horizontal shifts is key to predicting how a function's graph will look and behave based on its equation. Being able to anticipate this movement, along with understanding the parabolas and their vertices, empowers you to interpret and graph quadratic functions confidently.