Problem 26
Question
\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=(x+1)^{2} $$
Step-by-Step Solution
Verified Answer
Shift the parabola \( y = x^2 \) left by 1 unit to graph \( f(x) = (x+1)^2 \).
1Step 1: Identify the Standard Function
The standard function here is the basic quadratic function \( g(x) = x^2 \). This is a parabola that opens upwards with its vertex at the origin (0,0).
2Step 2: Apply the Horizontal Shift
The function \( f(x) = (x+1)^2 \) is derived from the standard function \( g(x) = x^2 \) by performing a horizontal shift. The term \( (x+1) \) indicates a shift to the left by 1 unit. So, we move the whole graph of \( g(x) = x^2 \) left by 1 unit.
3Step 3: Sketch the Transformed Graph
Starting from the standard parabola \( g(x) = x^2 \), apply the leftward shift by 1 unit to get the graph of \( f(x) = (x+1)^2 \). The vertex of the parabola moves from (0, 0) to (-1, 0), and the shape remains the same, opening upwards.
Key Concepts
Quadratic FunctionsHorizontal ShiftParabola
Quadratic Functions
A quadratic function is a fundamental concept in algebra and is represented by the mathematical formula \( f(x) = ax^2 + bx + c \). In its most basic form, the quadratic function is \( g(x) = x^2 \). This equation results in a U-shaped curve known as a parabola. Quadratic functions are one of the simplest types of polynomial functions, yet they hold great importance in various applications.
Quadratic functions can model a wide range of real-world scenarios, from projectile motion in physics to profit optimization in economics. The key feature of a quadratic function is its symmetry, centered around its vertex, and the ability it has to create parabolic graphs that either open upwards or downwards depending on the sign of \( a \).
When dealing with quadratic functions, the vertex form \( y = a(x - h)^2 + k \) is particularly useful for identifying transformations like horizontal and vertical shifts, as it highlights the vertex \((h, k)\) directly.
Quadratic functions can model a wide range of real-world scenarios, from projectile motion in physics to profit optimization in economics. The key feature of a quadratic function is its symmetry, centered around its vertex, and the ability it has to create parabolic graphs that either open upwards or downwards depending on the sign of \( a \).
When dealing with quadratic functions, the vertex form \( y = a(x - h)^2 + k \) is particularly useful for identifying transformations like horizontal and vertical shifts, as it highlights the vertex \((h, k)\) directly.
Horizontal Shift
Horizontal shifts are a form of graph transformation that move the graph of a function left or right along the x-axis. For quadratic functions such as \( f(x) = (x+1)^2 \), the horizontal shift can be identified by the modified structure of the input, here being \( (x+1) \).
When the function changes from \( g(x) = x^2 \) to \( f(x) = (x+1)^2 \), it signifies a leftward shift in the graph. This shift of 1 unit left is due to the addition inside the function \((+1)\), contrasting intuitive expectations that often predict moving right.
When the function changes from \( g(x) = x^2 \) to \( f(x) = (x+1)^2 \), it signifies a leftward shift in the graph. This shift of 1 unit left is due to the addition inside the function \((+1)\), contrasting intuitive expectations that often predict moving right.
- A function \( y = f(x-c) \) shifts right by \( c \) units, while \( y = f(x+c) \) shifts left by \( c \) units.
- Horizontal shifts do not affect the shape of the graph.
- It's only the position along the x-axis that alters, retaining the original parabola shape and direction.
Parabola
A parabola is the graph representation of a quadratic function and displays a distinct U-shaped curve. Depending on the function, this parabola can open upwards or downwards. For the basic quadratic function \( g(x) = x^2 \), the parabola opens upwards with the lowest point being the vertex at the origin \((0,0)\).
The parabola is symmetric, meaning that if you drew a vertical line through its vertex, both sides of the parabola mirror each other. This symmetry is a crucial property that helps in sketching and analyzing parabolas, as it simplifies calculations and understanding the graph.
When a parabola is shifted horizontally, as with the function \( f(x) = (x+1)^2 \), the vertex also moves but the symmetrical properties remain unchanged. Its vertex merely relocates to \((-1, 0)\), still opening upwards and maintaining a symmetric shape around the new vertex point.
The parabola is symmetric, meaning that if you drew a vertical line through its vertex, both sides of the parabola mirror each other. This symmetry is a crucial property that helps in sketching and analyzing parabolas, as it simplifies calculations and understanding the graph.
When a parabola is shifted horizontally, as with the function \( f(x) = (x+1)^2 \), the vertex also moves but the symmetrical properties remain unchanged. Its vertex merely relocates to \((-1, 0)\), still opening upwards and maintaining a symmetric shape around the new vertex point.
Other exercises in this chapter
Problem 26
Use the Inverse Function Property to show that \(f\) and \(g\) are inverses of each other. $$ f(x)=3 x, \quad g(x)=\frac{x}{3} $$
View solution Problem 26
Running Speed A man is running around a circular track that is \(200 \mathrm{~m}\) in circumference. An observer uses a stopwatch to record the runner's time at
View solution Problem 26
Sketch the graph of the function by first making a table of values. \(G(x)=|x|-x\)
View solution Problem 26
23- 30 . A function \(f\) is given. (a) Use a graphing device to draw the graph of \(f .\) (b) State approximately the intervals on which \(f\) is increasing an
View solution