Problem 56 PREREQUISITE SKILL Graph each pa... [FREE SOLUTION]

Chapter 14: Problem 56

PREREQUISITE SKILL Graph each pair of functions on the same set of axes. $$ y=2 x^{2}, y=2(x+1)^{2} $$

Short Answer

Expert verified

Both functions are parabolas; the second is the first shifted 1 unit left.

Step by step solution

Identify the Functions

The given functions are $ y = 2x^2 $ and $ y = 2(x+1)^2 $. These are both quadratic functions, which represent parabolas when graphed.

Determine the Vertex Form

The function $ y = 2x^2 $ is already in vertex form with the vertex at $(0,0)$ and opens upwards. The function $ y = 2(x+1)^2 $ is also in vertex form with the vertex at $(-1,0)$ and also opens upwards.

Choose the Graphing Window

Select a window size to properly see transformations. A typical window from $x=-5$ to $x=5$ and $y=-1$ to $y=10$ should be sufficient to see both functions clearly.

Graph the First Function

Plot the parabola $ y = 2x^2 $. This parabola opens upwards and is centered at the origin $(0,0)$. It becomes steeper as it moves away from the vertex.

Graph the Second Function

Plot the parabola $ y = 2(x+1)^2 $. This is the transformation of $ y = 2x^2 $ that shifts the graph 1 unit to the left, resulting in the vertex being at $(-1,0)$.

Compare the Graphs

Notice that $ y = 2(x+1)^2 $ is the same shape as $ y = 2x^2 $ but shifted horizontally left by 1 unit. Both have the same opening direction and width, as described by the coefficient of $x^2$.

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Key Concepts

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

Graphing Parabolas

Parabolas are a distinctive shape you often see in the graph of quadratic functions. When graphing a parabola, it's important to understand its main features, including the vertex, axis of symmetry, direction of opening, and width. The standard quadratic form is

The general equation of a parabola is given by \[ y = ax^2 + bx + c \]
If the leading coefficient $a$ is positive, the parabola opens upwards. If $a$ is negative, it opens downwards.
The vertex is either the maximum or minimum point of the parabola, depending on the direction it opens.

For instance, in the function $ y = 2x^2 $, the parabola opens upwards because the coefficient of $x^2$ is positive. The vertex, in this case, is at the origin $(0,0)$, showcasing where the curve changes direction. Choosing an appropriate window size, like

x-values ranging from $-5$ to $5$
y-values ranging from $-1$ to $10$

is crucial to capture the full curve and its transformations properly. This setup allows you to easily see how the parabola stretches, compresses, and shifts.

Vertex Form

The vertex form of a quadratic function offers a unique way to represent parabolas, emphasizing the vertex location and making function transformations clear. The vertex form is typically written as \[ y = a(x-h)^2 + k \], where:

$(h, k)$ is the vertex of the parabola, showing the point where it turns.
The value of $a$ determines the parabola's width and direction. A larger $a$ makes the parabola narrower; a smaller $a$ makes it wider.

Taking the function $y = 2(x+1)^2$, you notice the parabola in vertex form with

a vertex at $(-1, 0)$.
It signifies the parabola shifts horizontally to the left of the y-axis by 1 unit.
The same coefficient $a = 2$ indicates the parabola opens upwards with the same width as $ y = 2x^2 $.

Recognizing the vertex form helps in graphing parabolas accurately and understanding their transformations without having to perform extensive calculations.

Function Transformations

Function transformations involve shifting, stretching, or reflecting functions on a graph. Understanding these movements provides insight into how graphs relate across different functions or equations. Basic transformations include:

Horizontally shifting the graph is affected by the $h$ value in the vertex form $ y = a(x-h)^2 + k $. If $h$ is positive, the graph shifts right, and if negative, it shifts left.
Vertical shifts adjust the graph up or down based on the $k$ value.
Scaling or stretching is directed by the $a$ value, where values greater than 1 stretch the parabola, making it thinner.
Reflections occur when $a$ is negative, flipping the parabola over the x-axis.

For example, in $ y = 2(x+1)^2 $, the transformation manifests as a horizontal shift 1 unit to the left. The graph retains its steepness since the $a$ remains constant. Such transformations facilitate comparing two or more function graphs, like seeing that $ y = 2(x+1)^2 $ is a shifted version of $ y=2x^2 $ with identical widths and orientations.

91影视

PREREQUISITE SKILL Graph each pair of functions on the same set of axes. $$ y=2 x^{2}, y=2(x+1)^{2} $$

Short Answer

Step by step solution

Identify the Functions

Determine the Vertex Form

Choose the Graphing Window

Graph the First Function

Graph the Second Function

Compare the Graphs

Key Concepts

Graphing Parabolas

Vertex Form

Function Transformations

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