Accepted answer
Answer included
Problem
The keyword is “graph for y 5”. Interpreting this as the equation \(y = 5\), determine the graph on the coordinate plane and describe its key features.
Solution
1) Interpret the equation
The equation \(y = 5\) states that the \(y\)-coordinate is fixed at \(5\) for every possible \(x\)-value. This is a constant function.
2) Generate points that satisfy \(y = 5\)
Select any \(x\)-values and keep \(y\) equal to \(5\):
- If \(x = -4\), then the point is \((-4, 5)\).
- If \(x = 0\), then the point is \((0, 5)\).
- If \(x = 3\), then the point is \((3, 5)\).
All points \((x, 5)\) lie on the same horizontal level, so connecting them forms a horizontal line.
3) Identify slope and intercepts
Using two distinct points on the line, for example \((0,5)\) and \((2,5)\), the slope is
\[
m = \frac{\Delta y}{\Delta x} = \frac{5 - 5}{2 - 0} = \frac{0}{2} = 0.
\]
Intercepts:
- \(y\)-intercept: set \(x = 0\). Then \(y = 5\), so the line crosses at \((0,5)\).
- \(x\)-intercepts: set \(y = 0\). But \(y\) is always \(5\), so there are no \(x\)-intercepts.
4) Domain and range
Because \(x\) can be any real number while \(y\) remains fixed:
\[
\text{Domain} = (-\infty, \infty), \qquad \text{Range} = \{5\}.
\]
5) Summary table of features
| Feature | Value for \(y=5\) | Reason |
|---|---|---|
| Type | Constant function | \(y\) does not depend on \(x\) |
| Slope | \(m = 0\) | \(\Delta y = 0\) between any two points |
| \(y\)-intercept | \((0, 5)\) | At \(x=0\), \(y=5\) |
| \(x\)-intercepts | None | \(y\) never equals \(0\) |
| Domain | \((-\infty, \infty)\) | Any real \(x\) is allowed |
| Range | \(\{5\}\) | Only one output value occurs |
Visualization
Common checks
- Any point on the graph must have \(y=5\). For example, \((10,5)\) is on the line, while \((10,4)\) is not.
- If the line is horizontal, the slope must be \(0\); a nonzero slope indicates a different equation.
- No \(x\)-intercept exists because \(y\) never becomes \(0\).
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