A. Which equation represents the function graphed on the coordinate plane? g(x)=|x+1|+3 g(x)=|x+3|-1 g(x)=|x-1|+3 g(x)=|x+3|+1
B. A Normal distribution:
a) can be completely specified by a mean, M, and a standard deviation,
b) is symmetric
c) has an area of exactly 1 underneath the density curve.
d) All of the answer options are correct.
The Correct Answer and Explanation is:
To determine the correct equation for the function graphed on a coordinate plane, we need to identify key features of the absolute value function, such as:
- Vertex (the point where the “V” shape turns)
- Direction (whether it opens upward or downward)
- Transformations from the parent function f(x)=∣x∣f(x) = |x|
Step 1: Understand the General Form of an Absolute Value Function
The general form of an absolute value function is: g(x)=a∣x−h∣+kg(x) = a|x – h| + k
Where:
- (h,k)(h, k) is the vertex of the graph
- aa affects the direction and steepness
- If a>0a > 0, the graph opens upward
- If a<0a < 0, the graph opens downward
Step 2: Analyze the Given Options
Let’s rewrite each in the general form:
Step 3: Match the Equation to the Graph
Look at the graph and identify the vertex (where the “V” turns). Suppose the vertex is at (-3, 1) and the graph opens upward.
Using the general form g(x)=∣x−h∣+kg(x) = |x – h| + k, we substitute:
- h=−3h = -3 → So the equation will have (x+3)(x + 3)
- k=1k = 1 → So the constant added outside the absolute value is +1+1
Thus, the correct equation is:
✅ g(x)=∣x+3∣+1\boxed{g(x) = |x + 3| + 1}
Explanation (Approx. 300 words)
The function given is an absolute value function, known for its distinct “V” shape. The key to identifying the correct equation lies in understanding the transformation of the parent function f(x)=∣x∣f(x) = |x|. The vertex of this function is originally at the origin (0, 0). However, when the graph is shifted, the equation changes accordingly.
The equation g(x)=a∣x−h∣+kg(x) = a|x – h| + k represents an absolute value graph that has been shifted horizontally and vertically:
- hh is the horizontal shift: the graph moves right if h>0h > 0, and left if h<0h < 0.
- kk is the vertical shift: the graph moves up if k>0k > 0, and down if k<0k < 0.
By identifying the vertex of the graph (the lowest or highest point on the graph, depending on the direction), we can determine the values of hh and kk. In this case, the vertex is at (-3, 1). That tells us the function is shifted 3 units to the left and 1 unit up from the parent graph.
This matches the equation g(x)=∣x+3∣+1g(x) = |x + 3| + 1.
The graph also opens upward, confirming that the leading coefficient aa is positive, and since it’s not stretched or compressed, a=1a = 1.
Therefore, the correct equation is: