Which of the following is a point on the graph as described by the function y=(2x+1)2−4?(1 point)
Responses
(0,−2)
left parenthesis 0 comma negative 2 right parenthesis
(−1,−3)
left parenthesis negative 1 comma negative 3 right parenthesis
(1,2)
left parenthesis 1 comma 2 right parenthesis
(−1,−5)
To determine which of the given points lies on the graph of the function y = (2x + 1)^2 - 4, we can plug the x-coordinate of each point into the function and check if the resulting y-coordinate matches the one given in the point.
Let's test each point:
1. For the point (0,−2):
We substitute x with 0:
y = (2(0) + 1)^2 - 4
y = (1)^2 - 4
y = 1 - 4
y = -3
Since the y-coordinate is -3 instead of -2, point (0,−2) is not on the graph.
2. For the point (−1,−3):
We substitute x with -1:
y = (2(-1) + 1)^2 - 4
y = (-2 + 1)^2 - 4
y = (-1)^2 - 4
y = 1 - 4
y = -3
Since the y-coordinate is -3, point (−1,−3) is on the graph.
3. For the point (1,2):
We substitute x with 1:
y = (2(1) + 1)^2 - 4
y = (2 + 1)^2 - 4
y = (3)^2 - 4
y = 9 - 4
y = 5
Since the y-coordinate is 5 instead of 2, point (1,2) is not on the graph.
4. For the point (−1,−5):
We substitute x with -1 (this was already calculated when checking point (−1,−3)):
As we found earlier, the y-coordinate when x is -1 is -3, not -5. Thus, point (−1,−5) is not on the graph.
Therefore, the correct point on the graph is (−1,−3).
Which of the following is the graph of the function y=x+2?(1 point)
Responses
An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. A downward slanting line is graphed on the coordinate plane. The line passes through points plotted at left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 0 right parenthesis.
Image with alt text: An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. A downward slanting line is graphed on the coordinate plane. The line passes through points plotted at left parenthesis 0 comma 2 right parenthesis and left parenthesis 2 comma 0 right parenthesis.
An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 5 to 5 in one unit increments. A parabola opening upward is graphed on the coordinate plane. The parabola is formed by connecting points plotted at left parenthesis negative 4 comma 2 right parenthesis, left parenthesis negative 2 comma 0 right parenthesis, and left parenthesis 0 comma 2 right parenthesis.
Image with alt text: An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 5 to 5 in one unit increments. A parabola opening upward is graphed on the coordinate plane. The parabola is formed by connecting points plotted at left parenthesis negative 4 comma 2 right parenthesis, left parenthesis negative 2 comma 0 right parenthesis, and left parenthesis 0 comma 2 right parenthesis.
An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. A circle is graphed on the coordinate plane. The circle is formed by connecting points plotted at left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 0 comma 2 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.
Image with alt text: An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. A circle is graphed on the coordinate plane. The circle is formed by connecting points plotted at left parenthesis negative 2 comma 0 right parenthesis, left parenthesis 0 comma 2 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.
An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. An upward slanting line is graphed on the coordinate plane. The line passes through points plotted at left parenthesis negative 2 comma 0 right parenthesis, and left parenthesis 0 comma 2 right parenthesis.
Image with alt text: An illustration shows a coordinate plane with four quadrants. The x-axis and y-axis both range from negative 4 to 4 in one unit increments. An upward slanting line is graphed on the coordinate plane. The line passes through points plotted at left parenthesis negative 2 comma 0 right parenthesis, and left parenthesis 0 comma 2 right parenthesis.
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The function y = x + 2 is a linear equation in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept of the line.
From the equation:
- The slope (m) is 1 (since there's no coefficient written in front of x, it's understood to be 1).
- The y-intercept (b) is 2.
Now, let's analyze the given options based on our understanding of the graph of y = x + 2:
1. A downward slanting line through (0, 2) and (2, 0):
This graph cannot represent y = x + 2 because a line with a positive slope like 1 should be slanting upwards, not downwards.
2. A parabola opening upward:
This graph also cannot represent y = x + 2 because the function y = x + 2 is linear, not quadratic. A parabola is the graph of a quadratic function, not a linear one.
3. A circle:
Similarly, this graph cannot represent y = x + 2 because a circle is not the graph of a linear function. It's the shape for the equation of a circle, not a straight line.
4. An upward slanting line through (−2, 0) and (0, 2):
This graph represents a line with a positive slope, which matches the slope of the given function. It also crosses the y-axis at the y-intercept of 2, which corresponds to the "+2" in the function y = x + 2.
Therefore, the correct choice is the fourth one: an illustration showing a coordinate plane with an upward slanting line that passes through points (−2, 0) and (0, 2). This graph correctly illustrates the function y = x + 2.
Use the image to answer the question.
A line graph shows months of the year versus average rainfall. The x-axis represents month of the year, ranging from January to December. The y-axis shows the average rainfall in inches ranging from 0 to 4 in increments of 1. A curve is plotted on the graph that passes through the following approximate points left parenthesis January comma 2.3 right parenthesis, left parenthesis February comma 2 right parenthesis, left parenthesis March comma 2 right parenthesis, left parenthesis April comma 0.6 right parenthesis, left parenthesis May comma 0.6 right parenthesis, June comma 0.2 right parenthesis, left parenthesis July comma 0.45 right parenthesis, left parenthesis August comma 0.55 right parenthesis, left parenthesis September comma 0.7 right parenthesis, left parenthesis October comma 1.6 right parenthesis, left parenthesis November comma 2.4 right parenthesis, and left parenthesis December comma 3.2 right parenthesis.
Which of the following are the two quantities whose functional relationship is described in the given graph?
(1 point)
Responses
The two quantities are the average rainfall in inches and the month of the year.
The two quantities are the average rainfall in inches and the month of the year.
The two quantities are the average rainfall in inches and the years.
The two quantities are the average rainfall in inches and the years.
The two quantities are inches and months of the year.
The two quantities are inches and months of the year.
The two quantities are the x-values and the y-values.
The two quantities are the x -values and the y -values.
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The two quantities whose functional relationship is described in the given graph are:
The average rainfall in inches and the month of the year.
The graph plots the average rainfall (in inches) on the y-axis and correlates it with the month of the year on the x-axis. This is the standard format for representing this type of data, where time (in this case, months of the year) is plotted on the x-axis, and the variable being measured (average rainfall in inches) is plotted on the y-axis. The relationship shown in the graph is how the average rainfall changes over the different months of the year, which makes the two quantities:
1. The average rainfall in inches (the dependent variable, typically on the y-axis)
2. The month of the year (the independent variable, typically on the x-axis)
pick 1
The correct response is:
"The two quantities are the average rainfall in inches and the month of the year."
Use the image to answer the question.
A line graph shows months of the year versus average rainfall. The x-axis represents month of the year, ranging from January to December. The y-axis shows the average rainfall in inches ranging from 0 to 4 in increments of 1. A curve is plotted on the graph that passes through the following approximate points left parenthesis January comma 2.3 right parenthesis, left parenthesis February comma 2 right parenthesis, left parenthesis March comma 2 right parenthesis, left parenthesis April comma 0.6 right parenthesis, left parenthesis May comma 0.6 right parenthesis, June comma 0.2 right parenthesis, left parenthesis July comma 0.45 right parenthesis, left parenthesis August comma 0.55 right parenthesis, left parenthesis September comma 0.7 right parenthesis, left parenthesis October comma 1.6 right parenthesis, left parenthesis November comma 2.4 right parenthesis, and left parenthesis December comma 3.2 right parenthesis.
Consider the given graph. Analyze the intervals during the months from June to December and describe how that corresponds to the average rainfall.
(1 point)
Responses
the average rainfall increases then decreases
the average rainfall increases then decreases
the average rainfall decreases
the average rainfall decreases
the average rainfall increases
the average rainfall increases
the average rainfall remains constant
the average rainfall remains constant
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To analyze the intervals during the months from June to December in relation to the average rainfall as plotted on the graph, let's examine the approximate points given:
- June (0.2 inches)
- July (0.45 inches)
- August (0.55 inches)
- September (0.7 inches)
- October (1.6 inches)
- November (2.4 inches)
- December (3.2 inches)
Looking at these points, we can see a pattern in the average rainfall over these months. Starting in June, the average rainfall is at its lowest point at approximately 0.2 inches. From June onwards, the average rainfall increases each month:
- June to July: increases from 0.2 to 0.45 inches
- July to August: increases from 0.45 to 0.55 inches
- August to September: increases from 0.55 to 0.7 inches
- September to October: increases from 0.7 to 1.6 inches
- October to November: increases from 1.6 to 2.4 inches
- November to December: increases from 2.4 to 3.2 inches
There is no decrease in average rainfall from June through December; it consistently increases month by month.
Therefore, based on the information provided, the correct description of the interval from June to December in relation to average rainfall would be:
the average rainfall increases