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

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)

Answer 1

Based on the description you've provided of the piecewise function and its components, we can analyze whether the student's graph represents the function accurately. Given the function definition:

1. For \(x \leq -2\): The function is defined as \(f(x) = x + 3\). This is a linear function starting in Quadrant III with a positive slope. The piece ends at an open circle at \((-2, 1)\). Therefore, the graph should show this linear portion going upwards towards the point \((-2, 1)\) but not including it.

2. For \(-2 < x < 1\): The function is \(f(x) = x^2 - 1\), which is a quadratic function that opens upwards. The left endpoint at \(x = -2\) should show a closed circle at the point \((-2, 3)\), indicating that the function value at \(x = -2\) is indeed included in this piece. The vertex is located at \((0, -1)\) and the right endpoint of this portion is at \(x = 1\) with a closed circle at \((1, 0)\). So, the graph should form a parabola starting from \((-2, 3)\), reaching the vertex at \((0, -1)\), and then ending at \((1, 0)\).

3. For \(1 \leq x < 3\): The function is defined as \(f(x) = \log_2(-x + 3)\). This logarithmic function starts from an open circle at the point \((1, 2)\) and continues to the right, passing through the point \((5, 3)\). The graph should show this logarithm that increases and approaches but never reaches a vertical asymptote.

Evaluation of the Student's Graph

• Linear Portion: Verify that the linear function \(x + 3\) correctly ends with an open circle at \((-2, 1)\) and that it is drawn correctly with a positive slope.

• Quadratic Portion: Ensure that there is a closed circle at \((-2, 3)\) and that it smoothly transitions to the vertex at \((0, -1)\). Check that it makes a proper transition to the closed circle at \((1, 0)\) and correctly quits at \(x=1\).

• Logarithmic Portion: Make sure there is an open circle at the point \((1, 2)\) and that the function increases to the right, passing through \((5, 3)\) but not going past \(x=3\).

Steps for Correction

1. If the linear part contains a closed circle at \((-2, 1)\), it needs to be an open circle.
2. Check if the quadratic part has a closed circle at \((-2, 3)\) and a smooth, continuous curve representing the vertex correctly.
3. Ensure the logarithmic part starts with an open circle at \((1, 2)\) and correctly extends towards \(x = 3\).
4. Confirm that there is no overlap or discontinuity in the graph where the pieces of the piecewise function meet.

If any of these characteristics don't match between the student's graph and the function description, then the student's graph is not a correct representation. The specific corrections would depend on what errors were found in each of the three segments of the piecewise function.

Answer 2

You can only use words

Answer 3

The student's graph is not a correct representation of the piecewise function. Here are the necessary corrections:

1. Linear Portion: The graph should show a linear function \(f(x) = x + 3\) that extends to the point \((-2, 1)\) with an open circle, indicating that this point is not included in the domain of that piece. Ensure the line has a positive slope and begins in Quadrant III.

2. Quadratic Portion: The graph should depict the quadratic function \(f(x) = x^2 - 1\) beginning with a closed circle at \((-2, 3)\), meaning this point is included. The curve should reach the vertex at \((0, -1)\) and should end at a closed circle at \((1, 0)\), showing that the function value at \(x = 1\) is also included.

3. Logarithmic Portion: The graph must illustrate the logarithmic function \(f(x) = \log_2(-x + 3\) starting with an open circle at the point \((1, 2)\). This indicates that the value at \(x = 1\) does not belong to this piece. The function should increase to the right and pass through the point \((5, 3)\), while not extending beyond \(x = 3\).

To summarize, make sure the graph accurately represents proper endpoints (open or closed circles), transitions smoothly between the pieces, and aligns with the defined function segments.