Problem 16
Question
Graph the functions 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) \(y-\pi=(x-2 \pi)^{2}\) (b) \(y-\pi=-(x-2 \pi)^{2}\)
Step-by-Step Solution
Verified Answer
The transformed functions will result in different graphs. Function (a) will be a parabola facing upwards, shifted \(2\pi\) units to the right and \(\pi\) units up, with vertex at \((2 \pi, \pi)\). Function (b) will be a parabola facing downwards, shifted \(2\pi\) units to the right and \(\pi\) units up, with vertex at \((2\pi, \pi)\).
1Step 1: Identify and Understand the Transformations
For both functions, compare them with the basic function \(y = x^{2}\). Notice what has changed. In function (a), \(y-\pi=(x-2 \pi)^{2}\), it has been shifted \(\pi\) units upward and \(2\pi\) units to the right. For function (b), \(y-\pi=-(x-2 \pi)^{2}\), in addition to the shifts observed in function (a), it also has been flipped over the \(x\)-axis since there is a negative sign in front of the squared term.
2Step 2: Calculate the Critical Points
1. The \(x\)-intercept - set \(y=0\) and solve for \(x\): For both functions, \(x = 2\pi\). 2. The \(y\)-intercept - set \(x=0\) and solve for \(y\): For both functions, \(y = \pi + 4\pi^{2}\). 3. The vertices: For both functions, the vertex is at \((2\pi, \pi)\).
3Step 3: Graph the Functions
Plot the functions on a graph using the transformations and critical points we've identified. For function (a), start with the basic function \(y = x^{2}\), then shift it \(\pi\) units upward and \(2\pi\) units to the right. For function (b), do the same but also flip the graph over the \(x\)-axis.
Key Concepts
Function TransformationsQuadratic Function VertexX-Intercepts and Y-InterceptsReflection Across Axis
Function Transformations
Understanding function transformations is essential when graphing equations, especially quadratic functions. A function transformation involves shifting, reflecting, stretching, or compressing the graph of a base function, in this case, the parabola represented by the quadratic equation \( y = x^2 \).
Take, for instance, the functions \( y-\pi=(x-2\pi)^2 \) and \( y-\pi=-(x-2\pi)^2 \). In both cases, the presence of \( - \pi \) and \( 2\pi \) within the equation indicates transformations from the base function. The term \( y-\pi \) results in a vertical shift \( \pi \) units up, and \( x-2\pi \) results in a horizontal shift \( 2\pi \) units to the right.
For the second function, the negative sign in front of the squared term \( -(x-2\pi)^2 \) indicates an additional transformation, a reflection over the x-axis. This changes the direction the parabola opens. Recognizing these transformations allows us to predict the general shape and orientation of the graph before plotting any points.
Take, for instance, the functions \( y-\pi=(x-2\pi)^2 \) and \( y-\pi=-(x-2\pi)^2 \). In both cases, the presence of \( - \pi \) and \( 2\pi \) within the equation indicates transformations from the base function. The term \( y-\pi \) results in a vertical shift \( \pi \) units up, and \( x-2\pi \) results in a horizontal shift \( 2\pi \) units to the right.
For the second function, the negative sign in front of the squared term \( -(x-2\pi)^2 \) indicates an additional transformation, a reflection over the x-axis. This changes the direction the parabola opens. Recognizing these transformations allows us to predict the general shape and orientation of the graph before plotting any points.
Quadratic Function Vertex
The vertex of a quadratic function is a significant point on the graph where the direction of the function changes. It is the highest or lowest point on the graph, known as the maximum or minimum, respectively. In the standard form of a quadratic equation \( y = a(x-h)^2 + k \), the vertex is located at the coordinate \( (h, k) \).
In our examples, both functions have the same vertex at \( (2\pi, \pi) \). This is determined by looking at the transformations from the basic form \( y = x^2 \) to \( y - \pi = (x - 2\pi)^2 \), revealing that the vertex has been shifted to the right by \( 2\pi \) and up by \( \pi \). The vertex helps to anchor the graph, from which the parabola's shape is constructed upwards or downwards based on the value of \( a \).
In our examples, both functions have the same vertex at \( (2\pi, \pi) \). This is determined by looking at the transformations from the basic form \( y = x^2 \) to \( y - \pi = (x - 2\pi)^2 \), revealing that the vertex has been shifted to the right by \( 2\pi \) and up by \( \pi \). The vertex helps to anchor the graph, from which the parabola's shape is constructed upwards or downwards based on the value of \( a \).
X-Intercepts and Y-Intercepts
Identifying the x-intercepts and y-intercepts of a graph is crucial for a clear understanding of the function's behavior. An x-intercept is where the graph crosses the x-axis, and the y-intercept is where it crosses the y-axis. To find the x-intercepts of a quadratic function, set \( y = 0 \) and solve for \( x \). To find the y-intercepts, set \( x = 0 \) and solve for \( y \).
Looking at our example functions, the x-intercept for both is \( x = 2\pi \). This is calculated by setting y to zero. As for the y-intercepts, they are found by setting x to zero, resulting in \( y = \pi + 4\pi^2 \). It's important to note that while quadratic functions always have one y-intercept, they can have zero, one, or two x-intercepts depending on how the graph is positioned in relation to the x-axis.
Looking at our example functions, the x-intercept for both is \( x = 2\pi \). This is calculated by setting y to zero. As for the y-intercepts, they are found by setting x to zero, resulting in \( y = \pi + 4\pi^2 \). It's important to note that while quadratic functions always have one y-intercept, they can have zero, one, or two x-intercepts depending on how the graph is positioned in relation to the x-axis.
Reflection Across Axis
Reflection across an axis is a type of transformation that flips a graph over a specific axis. In the case of a quadratic function, a reflection across the x-axis will turn a parabola that opens upward to one that opens downward, or vice versa. This is due to multiplying the function by \( -1 \).
In our function \( y-\pi=-(x-2\pi)^{2} \), the negative sign before the squared term indicates that the graph of \( (x-2\pi)^{2} \) has been reflected across the x-axis. A reflection changes the direction of the parabola but retains the same vertex, line of symmetry, and x-intercepts. Understanding how reflection affects the graph aids in accurately sketching the parabola and visualizing its properties related to its orientation.
In our function \( y-\pi=-(x-2\pi)^{2} \), the negative sign before the squared term indicates that the graph of \( (x-2\pi)^{2} \) has been reflected across the x-axis. A reflection changes the direction of the parabola but retains the same vertex, line of symmetry, and x-intercepts. Understanding how reflection affects the graph aids in accurately sketching the parabola and visualizing its properties related to its orientation.
Other exercises in this chapter
Problem 15
Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answ
View solution Problem 16
How do the \(x\) - and \(y\) -intercepts of \(f(x)\) and \(g(x)\) affect the intercepts of \(f(x) g(x)\) ? State this as a general rule.
View solution Problem 16
Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answ
View solution Problem 17
Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -interc
View solution