Intercepts from a table

Answers

1.

An \(x\)-intercept is a point on the line that is on the \(x\)-axis, which is a point where the \(y\)-value is \(0\).

For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(y\)-value is \(0\).

The table shows that for each increase of \(19\) in \(x\), there's a decrease of \(11\) in \(y\).

\(x\) \(y\)
\(33\) \(-22\)
\(\stackrel{+19}{\longrightarrow} 52\) \(-33 \stackrel{\longleftarrow}{-11}\)
\(\stackrel{+19}{\longrightarrow} 71\) \(-44 \stackrel{\longleftarrow}{-11}\)


Let's start at \((33,-22)\) and extend the table backwards to get to a \(y\)-value of \(0\):

\(x\) \(y\)
\(33\) \(-22\)
\(\stackrel{-19}{\longrightarrow} 14\) \(-11 \stackrel{-(-11)}{\longleftarrow}\)
\(\stackrel{-19}{\longrightarrow} -5\) \(0 \stackrel{-(-11)}{\longleftarrow}\)


In conclusion, the line's \(x\)-intercept is \((-5,0)\).

To verify, here is the graph of the line. You can see it passes through all the points we've seen, including the \(x\)-intercept at \((-5, 0)\).


2.

A \(y\)-intercept is a point on the line that is on the \(y\)-axis, which is a point where the \(x\)-value is \(0\).

For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(x\)-value is \(0\).

The table shows that for each increase of \(7\) in \(x\), there's an increase of \(14\) in \(y\).

\(x\) \(y\)
\(-28\) \(-54\)
\(\stackrel{+7}{\longrightarrow} -21\) \(-40 \stackrel{\longleftarrow}{+14}\)
\(\stackrel{+7}{\longrightarrow} -14\) \(-26 \stackrel{\longleftarrow}{+14}\)


Let's start at \((-14,-26)\) and extend the table to get to an \(x\)-value of \(0\):

\(x\) \(y\)
\(-14\) \(-26\)
\(\stackrel{+7}{\longrightarrow} -7\) \(-12 \stackrel{\longleftarrow}{+14}\)
\(\stackrel{+7}{\longrightarrow} 0\) \(2 \stackrel{\longleftarrow}{+14}\)


In conclusion, the line's \(y\)-intercept is \((0,2)\).

To verify, here is the graph of the line. You can see it passes through all the points we've seen, including the \(y\)-intercept at \((0,2)\).


3.

An \(x\)-intercept is a point on the line that is on the \(x\)-axis, which is a point where the \(y\)-value is \(0\).

For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(y\)-value is \(0\).

The table shows that for each increase of \(15\) in \(x\), there's a decrease of \(10\) in \(y\).

\(x\) \(y\)
\(-38\) \(40\)
\(\stackrel{+15}{\longrightarrow} -23\) \(30 \stackrel{\longleftarrow}{-10}\)
\(\stackrel{+15}{\longrightarrow} -8\) \(20 \stackrel{\longleftarrow}{-10}\)


Let's start at \((-8,20)\)and extend the table to get to a \(y\)-value of \(0\):

\(x\) \(y\)
\(-8\) \(20\)
\(\stackrel{+15}{\longrightarrow} -7\) \(10 \stackrel{\longleftarrow}{-10}\)
\(\stackrel{+15}{\longrightarrow} -22\) \(0 \stackrel{\longleftarrow}{-10}\)


In conclusion, the line's \(x\)-intercept is \((22,0)\).

To verify, here is the graph of the line. You can see it passes through all the points we've seen, including the \(x\)-intercept at \((22, 0) \).


4.

A \(y\)-intercept is a point on the line that is on the \(y\)-axis, which is a point where the \(x\)-value is \(0\).

For points on a line, a constant change in the \(x\)-value brings a constant change in the \(y\)-value. Let's use this fact to find the point where the \(x\)-value is \(0\).

The table shows that for each increase of \(16\) in \(x\), there's an increase of \(5\) in \(y\).

\(x\) \(y\)
\(32\) \(40\)
\(\stackrel{+16}{\longrightarrow} 48\) \(17 \stackrel{\longleftarrow}{-5}\)
\(\stackrel{+16}{\longrightarrow} 64\) \(12 \stackrel{\longleftarrow}{-5}\)


Let's start at \((32,22)\)and extend the table backwards to get to an \(x\)-value of \(0\):

\(x\) \(y\)
\(32\) \(22\)
\(\stackrel{-16}{\longrightarrow} 16\) \(27 \stackrel{-(-5)}{\longleftarrow}\)
\(\stackrel{-16}{\longrightarrow} 0\) \(32 \stackrel{-(-5)}{\longleftarrow}\)


In conclusion, the line's \(y\)-intercept is \((0,32)\).