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)

Question

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.
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, 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.

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.
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Answer 1

Given the conditions of the function, we can derive its equation based on the information provided:

1. Initial Value (y-intercept): The function has an initial value of −1, which means the y-intercept (\(b\)) is \(-1\). Therefore, when \(x=0\), \(y=-1\).

2. Slope: The slope (\(m\)) is given as \(3\). This indicates that for every unit increase in \(x\), \(y\) increases by \(3\).

From this, we can construct the equation of the line in the slope-intercept form:

\[ y = mx + b \] \[ y = 3x - 1 \]

Finding the Intercepts

• Y-Intercept: As mentioned, when \(x = 0\), \(y = -1\). Therefore, the line intersects the y-axis at the point \((0, -1)\).

• X-Intercept: To find the x-intercept (\(y=0\)): \[ 0 = 3x - 1 \] \[ 3x = 1 \] \[ x = \frac{1}{3} \] So, the x-intercept is \(\left(\frac{1}{3}, 0\right)\).

Qualitative Features

• The function is a linear function that slopes upwards with a positive slope (3).
• It intersects the y-axis at the point (0, -1) and has a small positive x-intercept around \(0.33\).

Based on these criteria, the correct sketch of the graph should include:

• An upward line passing through the point (0, -1) on the y-axis.
• The x-intercept around \(\frac{1}{3}\), indicating where the line crosses the x-axis.
• The line will reside primarily in the first quadrant, but it will also cover parts of the fourth and third quadrants due to its upward slope.

Choosing the Correct Sketch

Among the sketches described:

• The one that matches these details is:
• An upward line with the correct intersection points for the x-axis and y-axis.

So, the correct option from the choices provided is the image showing a line intersecting the y-axis at (0, -1) and the x-axis at approximately (0.33, 0).

Answer 2

Which number was it?

Answer 3

The x-intercept of the function, which we derived to be \( \left(\frac{1}{3}, 0\right) \), is approximately \( 0.33 \).

So, the specific number you might be looking for in terms of the x-intercept is approximately 0.33.

Answer 4

Is it 1, 2, 3, or 4

Answer 5

Based on the provided descriptions for the sketches, the correct option that matches a function with an initial value of \(-1\) (y-intercept) and a slope of \(3\) (suggesting an upward line) is:

• The sketch that shows an upward line intersecting the y-axis at \((0, -1)\) and the x-axis at approximately \((0.3, 0)\).

Assuming the choices correspond to an earlier response in numbering format, the option you are describing is 4.