Properties of Functions Quick Check

Question

3 of 53 of 5 Items

Question
A function has an initial value of −1
and a slope equal to 3. Choose the sketch of a graph that exhibits the qualitative features of the function.(1 point)
Responses

A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows on both the ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma 3 right parenthesis.
Image with alt text: A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. A downward line with arrows on both the ends passes through the second, first, and fourth quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma 3 right parenthesis.

A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 0.3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis. Values are approximate.
Image with alt text: A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 0.3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis. Values are approximate.

A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis.
Image with alt text: A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, fourth, and third quadrants. The line intersects the x-axis at left parenthesis 3 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma negative 1 right parenthesis.

A coordinate plane with four quadrants shows the x and y axes ranging from negative 8 to 8 in increments of 1. An upward line with arrows on both the ends passes through the first, second, and third quadrants. The line intersects the x-axis at left parenthesis negative 1 comma 0 right parenthesis and the y-axis at left parenthesis 0 comma 3 right parenthesis.

Answer 1

To determine the correct sketch of the function based on the given information, we need to examine the key features:

• The initial value (y-intercept) is −1, which means the line crosses the y-axis at (0, -1).
• The slope is 3, which indicates that the line is increasing. For every unit increase in x, y increases by 3 units.

Based on these points:

1. The line must pass through the point (0, -1) on the y-axis.
2. The line must be rising (upward slope) as the slope is positive (3).
3. The line will extend through the first and third quadrants, as it rises upwards from the negative y-axis to the positive y-axis.

Now, let's analyze the provided options:

• First Response: The line is downward sloping (negative slope). Incorrect.
• Second Response: The line is upward sloping but intersects the x-axis at approximately (0.3, 0). This does not match since it also intersects the y-axis at (0, -1). Correct with respect to the y-intercept.
• Third Response: The line intersects the x-axis at (3, 0) and the y-axis at (0, -1). The line is upward sloping. This is still viable since it has the correct y-intercept, but the slope needs validation.
• Fourth Response: The line is upward sloping, intersects the x-axis at (-1, 0), and the y-axis at (0, 3). Incorrect.

The best match for the required features of the function, based on the initial conditions given, is the second response:

• The line is upward sloping.
• The y-intercept is correctly at (0, -1).

So, the correct sketch of the function is from the second response.