Question

lesson practice name problems 1–3 label each graph with a term from the word bank to describe the relationship within a circle that the graph represents. 1. radius diameter (m) 2. circumference diameter (m) 3. area diameter (m)

Explanation:

Step1: Analyze Graph 1 (Radius vs Diameter)

The relationship between radius \( r \) and diameter \( d \) of a circle is \( d = 2r \), which is a linear relationship. But the graph here is a curve? Wait, no—wait, radius on x - axis, diameter on y - axis. Wait, \( d = 2r \), so when \( r = 0 \), \( d = 0 \); \( r = 2 \), \( d = 4 \); \( r = 4 \), \( d = 8 \)? Wait, no, the graph shown: maybe mislabel? Wait, no, let's check the formulas for each:

- Radius - Diameter: \( d = 2r \) (linear, but the first graph is a curve? Wait, maybe the first graph is Area vs Diameter? No, the label is Radius (x - axis) and Diameter (y - axis). Wait, no, \( d = 2r \) is linear, so if the graph is a curve, maybe it's Area vs Radius? Wait, the problem says "label each y - axis with a term from the word bank to describe the relationship within a circle that the graph represents". The word bank (implied: Radius, Circumference, Area) and x - axis is Diameter.

Wait, let's recall the formulas:

1. Circumference \( C=\pi d\) (linear, \( C\) vs \( d\), slope \( \pi\))
2. Area \( A=\pi(\frac{d}{2})^2=\frac{\pi}{4}d^2\) (quadratic, curve)
3. Radius \( r=\frac{d}{2}\) (linear, \( r\) vs \( d\), slope \( \frac{1}{2}\))

Now, let's check each graph:

- Graph 2 (Circumference vs Diameter): The line is straight, so \( C = \pi d\) (linear), so this should be Circumference.
- Graph 3 (Area vs Diameter): Wait, no, the third graph is a straight line? No, wait the third graph (red) is a straight line? Wait, no, \( A=\frac{\pi}{4}d^2\) is a parabola (curve), but the first graph (blue) is a curve. Wait, maybe I mixed up. Wait, Graph 1 (blue, Radius label x - axis? No, x - axis is Radius? Wait, no, x - axis labels: first graph x - axis: 0,4,8,12,16,20 (maybe Radius), y - axis: Diameter. But \( d = 2r \), so it should be linear. But the graph is a curve. Wait, maybe the x - axis is Diameter and y - axis is Radius? No, \( r=\frac{d}{2}\) is linear. Wait, maybe the graphs are:

- Graph 1: y - axis should be Area (since \( A=\frac{\pi}{4}d^2\), curve), x - axis Diameter? But the label is Radius. Wait, no, the problem says "label each y - axis". So y - axis labels are Radius, Circumference, Area (the three on the right). So:

For each graph, x - axis is Diameter (m), y - axis is one of Radius, Circumference, Area.

1. Graph 1 (top, blue): y - axis label is Radius? No, wait the right - hand labels: first is Radius, second Circumference, third Area. So each graph's y - axis is to be labeled with Radius, Circumference, or Area, x - axis is Diameter.

So:

- Relationship 1: If y - axis is Area, \( A=\frac{\pi}{4}d^2\) (quadratic, curve) – matches the first graph (blue curve).
- Relationship 2: If y - axis is Circumference, \( C = \pi d\) (linear, straight line) – matches the second graph (green straight line).
- Relationship 3: If y - axis is Radius, \( r=\frac{d}{2}\) (linear, straight line) – but the third graph (red) is a straight line? Wait, no, the third graph (red) is a straight line with slope? Wait, \( r=\frac{d}{2}\), so when \( d = 0\), \( r = 0\); \( d = 4\), \( r = 2\); \( d = 8\), \( r = 4\). But the third graph's red line: when \( d = 0\), \( y = 0\); \( d = 4\), \( y = 2\)? Wait, no, the third graph's x - axis is Diameter (0,4,8,...) and y - axis is Area? No, the right - hand label for third is Area. Wait, I think I messed up. Let's re - express:

- Circumference vs Diameter: \( C=\pi d\), linear, so the graph with a straight line (Graph 2, green) should have y - axis labeled Circumference.
- Radius vs Diameter: \( r=\frac{d}{2}\), linear, so a straight line with slope \( \frac{1}{2}\).
- Area vs Diameter: \( A=\frac{\pi}{4}d^2\), quadratic, so a curve (Graph 1, blue) should have y - axis labeled Area? No, the first graph's right - hand label is Radius. Wait, the problem says "label each y - axis with a term from the word bank (Radius, Circumference, Area) to describe the relationship within a circle that the graph represents".

So:

1. Graph 1 (top, blue curve): The relationship is quadratic (since \( A=\frac{\pi}{4}d^2\) is quadratic), so this should be Area vs Diameter? But the right - hand label is Radius. Wait, no, maybe the x - axis is Radius and y - axis is Area: \( A=\pi r^2\), which is a curve (quadratic), so that matches. So if x - axis is Radius and y - axis is Area, then \( A=\pi r^2\) (curve), which matches the first graph.

2. Graph 2 (middle, green straight line): Circumference vs Diameter: \( C = \pi d\) (linear), so this is a straight line, matches.

3. Graph 3 (bottom, red straight line): Radius vs Diameter: \( r=\frac{d}{2}\) (linear), so this is a straight line, matches.

So to label the y - axes:

- Graph 1 (top): y - axis should be Area (since \( A=\pi r^2\), curve)
- Graph 2 (middle): y - axis should be Circumference (since \( C=\pi d\), linear)
- Graph 3 (bottom): y - axis should be Radius (since \( r=\frac{d}{2}\), linear)

Wait, but the problem says "label each y - axis with a term from the word bank to describe the relationship within a circle that the graph represents". The word bank terms are Radius, Circumference, Area (the three on the right, each next to a graph). So:

- Graph 1 (top, next to Radius label): No, the label on the right of Graph 1 is Radius, Graph 2 is Circumference, Graph 3 is Area. Wait, no, the right - hand side: first graph (top) has "Radius" to the right, second "Circumference", third "Area". So we need to check which relationship (Radius - Diameter, Circumference - Diameter, Area - Diameter) matches each graph.

- Radius - Diameter: \( r=\frac{d}{2}\) (linear, straight line)
- Circumference - Diameter: \( C=\pi d\) (linear, straight line, slope \( \pi\approx3.14\))
- Area - Diameter: \( A=\frac{\pi}{4}d^2\) (quadratic, curve)

Now, Graph 1 (top, blue curve): This is a curve, so it must be Area vs Diameter (since \( A\) vs \( d\) is quadratic). But the label next to it is Radius. So we need to assign the correct term to the y - axis. So:

- Graph 1 (y - axis label): Area (because the graph is a curve, matching \( A=\frac{\pi}{4}d^2\))
- Graph 2 (y - axis label): Circumference (because the graph is a straight line, matching \( C = \pi d\))
- Graph 3 (y - axis label): Radius (because the graph is a straight line, matching \( r=\frac{d}{2}\))

Wait, but the third graph (red) is a straight line. Let's check the slope: if x - axis is Diameter (d) and y - axis is Radius (r), then \( r=\frac{d}{2}\), so slope \( \frac{1}{2}\). If x - axis is Diameter and y - axis is Area, \( A=\frac{\pi}{4}d^2\) is curve. So:

- Graph 1 (blue curve): Area (y - axis)
- Graph 2 (green straight line): Circumference (y - axis)
- Graph 3 (red straight line): Radius (y - axis)

So the labels for the y - axes (from top to bottom) should be Area, Circumference, Radius? Wait, no, the original labels next to the graphs are Radius (top), Circumference (middle), Area (bottom). So we need to match the graph to the correct term:

- The curved graph (top) matches Area (since Area vs Diameter is quadratic, curve)
- The straight - line graph with steeper slope (middle, green) matches Circumference (since \( C=\pi d\), slope \( \pi\approx3.14\))
- The straight - line graph with gentle slope (bottom, red) matches Radius (since \( r=\frac{d}{2}\), slope \( \frac{1}{2}\))

So to label each y - axis:

1. Top graph (blue curve): y - axis label is Area (because \( A=\frac{\pi}{4}d^2\) is a quadratic relationship, curve)
2. Middle graph (green straight line): y - axis label is Circumference (because \( C = \pi d\) is a linear relationship, straight line)
3. Bottom graph (red straight line): y - axis label is Radius (because \( r=\frac{d}{2}\) is a linear relationship, straight line)

Step2: Confirm with Formulas

- Circumference formula: \( C=\pi d\) (linear, so straight line – matches middle graph)
- Area formula: \( A=\frac{\pi}{4}d^2\) (quadratic, so curve – matches top graph)
- Radius formula: \( r=\frac{d}{2}\) (linear, so straight line – matches bottom graph)

Answer:

• Graph 1 (top, blue curve): y - axis label is Area
• Graph 2 (middle, green straight line): y - axis label is Circumference
• Graph 3 (bottom, red straight line): y - axis label is Radius