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Chapter 8: Problem 72

Sketch the graph of each function. See Section 8.5. $$ f(x)=(x+1)^{2}+4 $$

Short Answer

Expert verified
The graph is a parabola with vertex at (-1, 4), opening upwards.

Step by step solution

01

Identify the Base Function

The base function is a simple quadratic function: \( f(x) = x^2 \). This is a parabola that opens upwards with its vertex at the origin (0,0).
02

Translate the Function Horizontally

The expression \( (x+1) \) inside the parentheses indicates a horizontal shift. The graph of the base function \( x^2 \) will shift 1 unit to the left, resulting in \( f(x) = (x+1)^2 \). The vertex of the shifted parabola is now at (-1, 0).
03

Translate the Function Vertically

The entire expression is added to 4, indicating a vertical shift upward by 4 units. Therefore, the vertex of the graph of \( f(x) = (x+1)^2 + 4 \) moves to (-1, 4).
04

Sketch the Graph

Using these transformations, sketch a curve that opens upwards, has its vertex at (-1, 4), and maintains the same parabolic shape as \( x^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Transformations
When dealing with quadratic functions, understanding quadratic transformations is essential for graphing. The basic quadratic function is given by \( f(x) = x^2 \), which is a parabola opening upwards. Transformations can shift, stretch, compress, or reflect this simple shape in various ways.
  • Horizontal Shifts: This occurs when there's a change inside the parenthesis of the squared term, such as \( f(x) = (x + 1)^2 \). This indicates a horizontal shift of the graph. If a value is added to \( x \), the graph shifts to the left. If subtracted, it shifts to the right.
  • Vertical Shifts: Changes outside the squared term, like the \(+4\) in \( f(x) = (x + 1)^2 + 4 \), dictate vertical movements of the graph. Adding a positive number shifts the graph upwards, while a negative number shifts it downwards.
These transformations do not change the shape of the parabola; it remains symmetrical and faces upwards unless a reflection transformation is applied.
Parabola Vertex
The vertex of a parabola is a crucial point which tells us the maximum or minimum value of a quadratic function, depending on the parabola's orientation. For the standard quadratic function \( f(x) = x^2 \), the vertex is at the origin (0,0). In our example \( f(x) = (x + 1)^2 + 4 \), the vertex moves to (-1, 4) due to transformations.
  • Finding the Vertex: The vertex is found at the point \((h, k)\), where \(h\) and \(k\) come from rewriting the quadratic function in the form \( f(x) = (x - h)^2 + k \).
  • Interpreting Vertex Form: The coordinates \((h, k)\) describe the location of the vertex of the transformed parabola. Here, \(-1\) and \(4\) indicate a leftward shift by 1 and an upward shift by 4, respectively.
Having a clear understanding of how to identify and plot the vertex helps in sketching an accurate graph of the quadratic function.
Horizontal and Vertical Shifts
Horizontal and vertical shifts are fundamental transformation operations applied to functions. They are critical in accurately sketching quadratic functions like our example \( f(x) = (x + 1)^2 + 4 \).
  • Horizontal Shift: When a transformation, such as \( (x + 1) \), appears, the graph shifts to the left by the absolute value of the addition, in this case, 1 unit left. It may seem counterintuitive, but adding to \( x \) results in a left shift.
  • Vertical Shift: This shift occurs due to the constant outside the square, as with \(+ 4\). This moves the whole graph of the function vertically up by 4 units, moving features like the vertex accordingly.
These shifts modify the graph's positions without altering its shape, offering a straightforward way to model real-world phenomena using quadratic functions.

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