Question

Which statement holds true for absolute value functions? A) The absolute value determines the direction in which the graph opens. B) The coefficient determines the line along which the graph is symmetrical. C) The distance between the left and the right arm is based on the coordinates of the vertex. D) The vertex coordinates and the absolute value determine the region of the graph on the plane.

Answer 1

The correct answer is:
D) the vertex coordinates and the absolute value determine the region of the graph on the plane.

Explanation:

The vertex coordinates of an absolute value function determine how far the graph is shifted vertically and horizontally from the origin, as the vertex of the parent graph y=|x| is the origin (0, 0).

The coefficient in front of the absolute value bars determines which direction the graph opens up in as well as how wide or narrow the bars are (the distance between the arms).  If the coefficient is positive, the graph opens upward; if the coefficient is negative, the graph opens downward.  If the coefficient is an integer greater than 1 or less than -1, then the graph will be more narrow than the parent graph; if the coefficient is a fraction, then the graph will be wider than the parent graph.

Since the vertex affects the position and the coefficient determines the direction it opens up, these determine the region of the graph on the plane.

Answer 2

The correct option is $\boxed{\bf option (D)}$.

Further explanation:

Absolute value function:

An absolute function is a function which always gives a positive for any real value of domain.  

Absolute value of a number is distance from 0 on the number line.

Example:

The absolute value function is expressed as shown below.

$f(x)=|x|=\begin{cases}x\ \ , & x\ge0\\-x, &x$

Concept used:

If a constant is added to the absolute value function, the graph of the function shifts vertically upwards if the constant is positive and it shifts vertically downward if the constant is negative.

The graph of $f(x)=|x|+a$ is shifted vertically upwards and the graph of $f(x)=|x|-a$ shift vertically downwards by $a$ units.

Here, $a$ is a constant.

Consider that absolute value function is in the form as $y=b|x|$.

If the value of $b$ is positive then the curve of the function open upwards and if the value of $b$ is negative then the curve of the function open downwards.

If the value of $b$ greater than $1$ or less than $-1$ then the graph becomes narrower.

The graph of $f(x)=|2x|$ is narrower.

If the coefficient of the absolute value function is in fraction then the graph of the function becomes wider.

The graph of the function $f(x)=|\frac{x}{2}|$ is wider.

Calculation:

The first option is incorrect because the absolute value function cannot determine the direction in which graph opens it is decided by the sign of the function.

The graph of the function $f(x)=-|x|$ is open downward and graph of $f(x)=|x|$ the function $f(x)=|x|$ opens upward.

The second option is incorrect because the coefficient cannot determine the symmetry of the graph.

The graph of the function $f(x)=|x|$ is symmetric about $y$-axis and the graph of the function $f(x)=-|x|$ is also symmetric about $y$-axis.

The third option is incorrect because the distance between right arm and left arm depends upon the coefficient of the absolute value function and if it is fraction then the graph becomes wider and if it is integer and greater than one than the graph becomes narrower.

The graph of $f(x)=|\frac{x}{2}|$ is wider and graph of  $f(x)=|2x|$ is narrower than graph of $f(x)=|\frac{x}{2}|$.

The fourth option is correct. From figure 1 (attached in the end) it is observed that the vertex coordinates and the absolute value determines the region of the graph on the plane.

This implies that correct option is option (D).

Therefore, the correct option is $\boxed{\bf option (D)}$.

Learn more:

1. Coordinate of the point : brainly.com/question/1286775

2. Equation: brainly.com/question/1473992

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Function

Keywords:  Coordinate geometry, x-axis, y-axis, x-coordinate, y-coordinate, Absolute value function, graph , region, coefficient, vertically opens, horizontally downs.