Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Graph the functions in Problems 10 through 18 by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=(x-1)^{2}\) (b) \(y=-x^{2}-1\)

Step-by-Step Solution

Answer

(a) Vertex: (1,0), Y-intercept: (0,1), X-intercept: (1,0). (b) Vertex: (0,-1), Y-intercept: (0,-1), no X-intercept

Step 1 Identifying the basic form of the function

The given function in (a) \(y=(x-1)^{2}\) is a quadratic function in standard form \(y=a(x-h)^{2}+k\). Here, \(a=1\), \(h=1\) and \(k=0\). In function (b) \(y=-x^{2}-1\), the \(a=-1\), \(h=0\) and \(k=-1\).

Step 2 Determine the Vertex

The vertex of function (a) will be at the point (h, k) which is (1, 0). For function (b), the vertex will be at point (0,-1).

Step 3 Calculate the y-intercepts

The y-intercepts can be found by setting x = 0. So for function (a), when \(x = 0, y = (0 - 1)^{2} = 1\). Therefore, the y-intercept is at point (0,1). For function (b), when \(x = 0, y = -0^{2}-1 = -1\). Therefore, the y-intercept is at point (0,-1).

Step 4 Calculate the x-intercepts

The x-intercepts can be found by setting y = 0. For function (a), when \(y = 0, x = 1\). Therefore, the x-intercept is at point (1,0). For function (b), there are no x-intercepts because y does not equal zero for any value of x

Step 5 Graphing the functions

The function (a) is graphed by drawing a parabolic curve opening upwards with the vertex at point (1,0), y-intercept at (0,1) and the x-intercept at (1,0). Function (b) is graphed by drawing a parabolic curve opening downwards with vertex at point (0,-1) and y-intercept at (0,-1). There is no x-intercept in function (b).

Key Concepts

Function Transformation

Understanding how a function is transformed helps you easily graph it. Transformations can include shifts, stretches, compressions, and reflections.
For quadratic functions, these transformations change the shape and position of the parabola:

• Shifts: The graph can move horizontally or vertically. For example, in function (a) \(y = (x-1)^2\), subtracting 1 from \(x\) shifts the parabola to the right by 1 unit.
• Reflections: A negative sign in front of the squared term, as in function (b) \(y = -x^2 - 1\), flips the parabola vertically.
• Stretches and Compressions: If \(a\) is greater than 1, the parabola becomes narrower. If \(a\) is between 0 and 1, it becomes wider.

These transformations allow you to predict and understand the behavior of the graph effectively.

Graphing Parabolas

Graphing a quadratic function involves sketching its parabolic shape. This is done by plotting key features like vertices, intercepts, and direction of opening.
To graph functions like those given, follow these steps:

• Identify the vertex. It's a critical point where the parabola either peaks or troughs. In function (a), the vertex is \((1,0)\). In function (b), it's \((0,-1)\).
• Determine the direction: If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards, like in function (b).
• Find intercepts: These points provide additional guidance for sketching. The \(y\)-intercepts are (0,1) and (0,-1) for functions (a) and (b) respectively. Function (a) has an \(x\)-intercept at (1,0), while function (b) has none.

Once outlined, connect the dots to form the smooth, U-shaped curve characteristic of a parabola.

Vertex of a Parabola

The vertex is the highest or lowest point on a parabola, depending on its direction. It provides a pivotal reference point for graphing.
For quadratic functions in the form \(y = a(x-h)^2 + k\), the vertex is at \((h, k)\).

• In function (a) \(y = (x-1)^2\), the vertex is \((1, 0)\). This is found by identifying \(h = 1\) and \(k = 0\).
• In function (b) \(y = -x^2 - 1\), the vertex is \((0, -1)\), where \(h = 0\) and \(k = -1\).

The position of the vertex helps determine the symmetry of the parabola and how transformations affect its graph. Knowing the vertex allows for accurate sketches and a better understanding of the function's behavior.