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Chapter 0: Problem 55

Use transformations to graph each function. $$ y=-2(x+3)^{2}-4 $$

Short Answer

Expert verified
Shift left 3 units, stretch by 2, reflect across x-axis, and shift down 4.

Step by step solution

01

Identify the parent function

The parent function is the basic quadratic function: \( y = x^2 \).
02

Horizontal shift of the graph

The equation \( (x+3) \) inside the parentheses indicates a horizontal shift. The graph shifts left by 3 units.
03

Vertical stretch and reflection

The coefficient \(-2\) indicates a vertical stretch by a factor of 2 and a reflection across the x-axis.
04

Vertical shift of the graph

The constant term \(-4\) outside the parentheses indicates a vertical shift. The graph shifts down by 4 units.
05

Combine transformations

Combine all the transformations: Start from the parent function \( y = x^2 \). Shift the graph left by 3 units, stretch it vertically by a factor of 2, reflect it across the x-axis, and shift it down by 4 units.

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Key Concepts

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

Quadratic Functions
Quadratic functions are polynomial functions of degree 2, typically in the form: \[ y = ax^2 + bx + c \] The graph of a quadratic function is a U-shaped curve called a parabola. The most basic quadratic function, known as the parent function, is: \[ y = x^2 \] In this form, the vertex of the parabola is at the origin (0,0), and the parabola opens upward. Transformations can shift and reshape the graph, changing its position and appearance.
Horizontal Shift
A horizontal shift occurs when the graph moves left or right along the x-axis. For the function \[ y = -2(x+3)^{2}-4 \] the term \[(x+3)\] inside the parentheses indicates a horizontal shift. Here, the graph moves left by 3 units. Remember that adding to the x-value inside the function shifts the graph left, whereas subtracting would shift it right.
Vertical Stretch
A vertical stretch changes the steepness of the graph. For \[ y = -2(x+3)^{2}-4 \] we have a coefficient of \[-2\] which first indicates a reflection (explained in the next section), and then a vertical stretch by a factor of 2. This means the graph is twice as steep compared to the parent function \[ y = x^2 \]. It stretches away from the x-axis. If the coefficient was a fraction between 0 and 1, it would compress the graph.
Reflection
A reflection flips the graph over a given axis. Here, in \[ y = -2(x+3)^{2}-4 \] the negative sign in front of the 2 indicates a reflection across the x-axis. This flips the parabola upside down. So instead of opening upwards, the graph opens downwards. The rest of the transformations still apply to this reflected graph.
Vertical Shift
A vertical shift moves the graph up or down along the y-axis. For \[ y = -2(x+3)^{2}-4 \] the term \[-4\] outside the parentheses indicates a vertical shift. This moves the entire graph down by 4 units. If it was a positive number, the graph would move up.

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