Feature

Question

Feature
The graph of a piecewise function is shown.The graph of a linear function starts in quadrant 3, has a positive slope, and a right endpoint at open circle left-parenthesis negative 2 comma 1 right parenthesis. It also shows a portion of a quadratic function opening upwards with left endpoint closed circle left-parenthesis negative 2 comma 3 right-parenthesis, vertex left-parenthesis 0 comma negative 1 right-parenthesis, and right endpoint closed circle left-parenthesis 1 comma 0 right-parenthesis. It also shows the graph of a logarithmic function with left endpoint open circle left-parenthesis 1 comma 2 right-parenthesis. The logarithmic function increases as it goes to the right and passes through the point left-parenthesis 5 comma 3 right parenthesis.
Question
Use the graph to answer the question.
A student was given the piecewise functionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row x plus 3 comma x less than or equal negative 2 2nd row x squared minus 1 comma negative 2 less than x less than 1 3rd row log subscript 2 baseline left parenthesis negative x plus 3 right parenthesis comma 1 less than or equals x less than 3 end layout and created this graph of the function.

Is the student’s graph a correct representation of the function? If not, explain how it should be corrected.
(3 points)
 
Skip to navigation

GPT-4o mini · Answer

To determine if the student's graph of the piecewise function $f (x)$ is a correct representation, we need to analyze each piece of the function according to the defined intervals.

The piecewise function is defined as follows:

For $x \leq - 2$ : $f (x) = x + 3$
- This is a linear function with a slope of 1. The graph starts in quadrant 3 and, at $x = - 2$ , should yield $f (- 2) = - 2 + 3 = 1$ . This is represented by an open circle at $(- 2, 1)$ .
For $- 2 < x < 1$ : $f (x) = x^{2} - 1$
- This is a quadratic function that opens upwards. At $x = - 2$ , $f (- 2) = (- 2)^{2} - 1 = 4 - 1 = 3$ , which corresponds to a closed circle at $(- 2, 3)$ . The vertex is located at $(0, - 1)$ and the value at $x = 1$ is $f (1) = 1^{2} - 1 = 0$ , which corresponds to a closed circle at $(1, 0)$ .
For $1 \leq x < 3$ : $f (x) = \log_{2} (- x + 3)$
- The logarithmic function will be evaluated starting at $x = 1$ , where $f (1) = \log_{2} (- 1 + 3) = \log_{2} (2) = 1$ . This part of the function has an open circle at $(1, 2)$ and increases, passing through the point $(5, 3)$ , which is consistent with logarithmic behavior.

Analysis of the Student's Graph

The linear part should start in quadrant 3 and show an open circle at $(- 2, 1)$ . It should have a 1 in the y-coordinate at $(- 2, 1)$ and extend downward to the left.
The quadratic part should start at $(- 2, 3)$ (closed circle) and go through the vertex at $(0, - 1)$ and end at $(1, 0)$ (closed circle).
The logarithmic part should start at $(1, 2)$ (open circle), and then it should increase, passing through $(5, 3)$ .

Corrections Needed

The graph of the linear function needs to be adjusted to reflect that it ends with an open circle at $(- 2, 1)$ , not a closed circle.
The quadratic part should correctly start from a closed circle at $(- 2, 3)$ , pass through the vertex, and end at $(1, 0)$ with a closed circle.
Ensure the logarithmic part starts at $(1, 2)$ (open circle) and behaves as described above.

If these characteristics are correctly updated in the student’s graph, it will accurately represent the given piecewise function $f (x)$ .