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Chapter 1: Problem 57

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ f(x)=4(x+1)^{2}-5 $$

Short Answer

Expert verified
A parabola shifted left 1 unit, stretched vertically by 4, then moved down 5 units.

Step by step solution

01

Identify the Toolkit Function

The given function is \( f(x) = 4(x+1)^2 - 5 \). The basic toolkit function here is \( y = x^2 \), which is a standard quadratic function represented by a parabola that opens upwards with its vertex at the origin \((0,0)\).
02

Horizontal Shift

Next, observe the transformation inside the parentheses: \((x + 1)^2\). This indicates a horizontal shift. The graph will shift 1 unit to the left because of \((x + 1)\).
03

Vertical Stretch

The coefficient 4 in the expression \(4(x+1)^2\) applies a vertical stretch to the graph of the function. The parabola becomes narrower as the graph is stretched vertically by a factor of 4.
04

Vertical Shift

The term \(-5\) at the end of the equation \(4(x+1)^2 - 5\) represents a vertical shift downward by 5 units. This will move the vertex of the parabola down by 5 units.
05

Vertex Form and Graph Sketch

Combining all transformations, the vertex form \( f(x) = a(x-h)^2 + k \) will have the vertex at \((-1, -5)\). The graph is a parabola opening upwards that is narrow, shifted left by 1 unit, and down by 5 units. Sketch this new graph based on the movements described.

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Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial function that is represented in its standard form as \( y = ax^2 + bx + c \). It is called quadratic because its highest degree term is a square (\( x^2 \)).
  • The graph of a quadratic function is a curve called a parabola.
  • It usually takes the shape of a "U" and can open upwards or downwards.
  • This curve is symmetric around the vertical line that passes through its vertex.
The vertex of the parabola represents the maximum or minimum point. In an upward opening parabola, the vertex is the minimum point, and in a downward opening one, it is the maximum point. The general form \( y = ax^2 \) serves as the "toolkit" quadratic function and is a perfect representation of a general parabolic curve.
Horizontal Shift
The concept of horizontal shifts allows us to move the entire graph of a function left or right on the coordinate plane. In the given function \( f(x) = 4(x+1)^2 - 5 \), the term inside the parentheses \((x + 1)\) produces this horizontal shift.
  • To determine the direction of the shift, observe the sign inside the parentheses.
  • If the expression is \((x - h)\), the shift is to the right by \(h\) units.
  • If the expression is \((x + h)\), the shift is to the left by \(h\) units.
For our function, the \((x+1)\) indicates a leftward shift by 1 unit. Such shifts are fundamental to function transformations and are pivotal in changing the position of the graph without affecting its shape.
Vertical Stretch
Vertical stretching refers to the expansion or contraction of a graph in the vertical direction. When we observe a coefficient other than 1 multiplying the square term as in \( 4(x+1)^2 \), it results in a vertical stretch. In this particular formula:
  • The coefficient 4 means that every point on the parabola is stretched vertically away from the x-axis by a factor of 4.
  • Vertical stretches result in the parabola appearing narrower compared to the standard form.
Vertical stretching helps in understanding modifying parameters in quadratic functions, influencing how "steep" or "shallow" the parabola appears. This is crucial when predicting the physical manifestations of quadratic scenarios, such as the path of projectiles or modeling quadratic trends.
Parabola
A parabola is a symmetrical open plane curve which is the graph of a quadratic function. It is one of the most recognizable shapes in mathematics due to its consistent properties and applications.
  • Parabolas can have different orientations; they can open upwards, downwards, leftwards, or rightwards, based on the function.
  • The most common form in quadratic functions is an upward or downward opening.
The location and direction of a parabola provide important information, such as focusing properties that are used in real-world applications like satellite dishes and headlights. By understanding its symmetries, vertices, and orientation, we can better analyze any quadratic function and properly sketch or model it in various contexts.

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Most popular questions from this chapter

Select all of the following graphs which represent \(y\) as a function of \(x\).

Determine the interval(s) on which the function is concave up and concave down. $$ b(x)=\sqrt[3]{-x-6} $$

For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down. $$ \begin{array}{|l|l|} \hline x & g(x) \\ \hline 1 & -200 \\ \hline 2 & -190 \\ \hline 3 & -160 \\ \hline 4 & -100 \\ \hline 5 & 0 \\ \hline \end{array} $$

Sketch a graph of each function as a transformation of a toolkit function. $$ k(x)=(x-2)^{3}-1 $$

Determine the interval(s) on which the function is concave up and concave down. $$ k(x)=-3 \sqrt{x}-1 $$
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