Question

enhanced algebra concepts & connections unit 1
georgias k - 12 mathematics standards
name:
date:
key features of linear functions
describe each function graphed above based on the key features requested. the first one has
been completed for you
graph 1: ( y = 2x - 4 )
intercepts (ordered pairs)
y - intercept ( (0,-4) )
x - intercept ( (2,0) )
slope
positive ( (y>0) )
( x>2 )
negative ( (y < 0) )
( x < 2 )
domain
d: ( { x | x in r } ) (set of all real numbers)
range
r: ( { y | y in r } )
end behavior
as ( x \to +infty, f(x) \to +infty )
as ( x \to -infty, f(x) \to -infty )
graph 2: ( y=-2x + 2 )
intercepts (ordered pairs)
y - intercept
x - intercept
slope
positive ( (y>0) )
negative ( (y < 0) )
domain
range
end behavior
graph 3: ( y=\frac{1}{2}x - 5 )
intercepts (ordered pairs)
y - intercept
x - intercept
slope
positive ( (y>0) )
negative ( (y < 0) )
domain
graph 4: ( x + y = 3 )
intercepts (ordered pairs)
y - intercept
x - intercept
slope
positive ( (y>0) )
negative ( (y < 0) )
domain
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page 3 of 10

Explanation:

Graph 2: \(y = -2x + 2\)

Intercepts (ordered pairs)

Step1: Find \(y\)-intercept

Set \(x = 0\) in \(y=-2x + 2\).
\(y=-2(0)+2=2\). So \(y\)-intercept is \((0,2)\).

Step2: Find \(x\)-intercept

Set \(y = 0\) in \(y=-2x + 2\).
\(0=-2x + 2\), then \(2x=2\), \(x = 1\). So \(x\)-intercept is \((1,0)\).

Slope

The equation is in \(y=mx + b\) form (\(m\) is slope). Here \(m=-2\).

Positive (\(y>0\))

Solve \(-2x + 2>0\).
\(-2x>-2\), divide both sides by \(- 2\) (reverse inequality sign), \(x < 1\).

Negative (\(y<0\))

Solve \(-2x + 2<0\).
\(-2x<-2\), divide both sides by \(-2\) (reverse inequality sign), \(x>1\).

Domain

For linear function \(y=-2x + 2\), \(x\) can be any real number. So \(D:\{x|x\in R\}\).

Range

For linear function \(y=-2x + 2\), \(y\) can be any real number. So \(R:\{y|y\in R\}\).

End Behavior

As \(x\to+\infty\), \(y=-2x + 2\to-\infty\).
As \(x\to-\infty\), \(y=-2x + 2\to+\infty\).

Graph 3: \(y=\frac{1}{2}x - 5\)

Intercepts (ordered pairs)

Step1: Find \(y\)-intercept

Set \(x = 0\) in \(y=\frac{1}{2}x-5\).
\(y=\frac{1}{2}(0)-5=-5\). So \(y\)-intercept is \((0, - 5)\).

Step2: Find \(x\)-intercept

Set \(y = 0\) in \(y=\frac{1}{2}x-5\).
\(0=\frac{1}{2}x-5\), then \(\frac{1}{2}x=5\), \(x = 10\). So \(x\)-intercept is \((10,0)\).

Slope

The equation is in \(y = mx + b\) form. Here \(m=\frac{1}{2}\).

Positive (\(y>0\))

Solve \(\frac{1}{2}x-5>0\).
\(\frac{1}{2}x>5\), \(x>10\).

Negative (\(y<0\))

Solve \(\frac{1}{2}x-5<0\).
\(\frac{1}{2}x<5\), \(x<10\).

Domain

For linear function \(y=\frac{1}{2}x - 5\), \(x\) can be any real number. So \(D:\{x|x\in R\}\).

Range

For linear function \(y=\frac{1}{2}x - 5\), \(y\) can be any real number. So \(R:\{y|y\in R\}\).

End Behavior

As \(x\to+\infty\), \(y=\frac{1}{2}x - 5\to+\infty\).
As \(x\to-\infty\), \(y=\frac{1}{2}x - 5\to-\infty\).

Graph 4: \(x + y=3\) (rewrite as \(y=-x + 3\))

Intercepts (ordered pairs)

Step1: Find \(y\)-intercept

Set \(x = 0\) in \(y=-x + 3\).
\(y=-0 + 3=3\). So \(y\)-intercept is \((0,3)\).

Step2: Find \(x\)-intercept

Set \(y = 0\) in \(y=-x + 3\).
\(0=-x + 3\), \(x = 3\). So \(x\)-intercept is \((3,0)\).

Slope

The equation \(y=-x + 3\) is in \(y=mx + b\) form. Here \(m=-1\).

Positive (\(y>0\))

Solve \(-x + 3>0\).
\(-x>-3\), \(x<3\).

Negative (\(y<0\))

Solve \(-x + 3<0\).
\(-x<-3\), \(x>3\).

Domain

For linear function \(y=-x + 3\), \(x\) can be any real number. So \(D:\{x|x\in R\}\).

Range

For linear function \(y=-x + 3\), \(y\) can be any real number. So \(R:\{y|y\in R\}\).

End Behavior

As \(x\to+\infty\), \(y=-x + 3\to-\infty\).
As \(x\to-\infty\), \(y=-x + 3\to+\infty\).

Answer:

• Graph 2:
• Intercepts: \(y\)-intercept \((0,2)\), \(x\)-intercept \((1,0)\)
• Slope: \(-2\)
• Positive (\(y > 0\)): \(x<1\)
• Negative (\(y < 0\)): \(x>1\)
• Domain: \(D:\{x|x\in R\}\)
• Range: \(R:\{y|y\in R\}\)
• End - Behavior: As \(x\to+\infty,y\to-\infty\); As \(x\to-\infty,y\to+\infty\)
• Graph 3:
• Intercepts: \(y\)-intercept \((0,-5)\), \(x\)-intercept \((10,0)\)
• Slope: \(\frac{1}{2}\)
• Positive (\(y > 0\)): \(x>10\)
• Negative (\(y < 0\)): \(x<10\)
• Domain: \(D:\{x|x\in R\}\)
• Range: \(R:\{y|y\in R\}\)
• End - Behavior: As \(x\to+\infty,y\to+\infty\); As \(x\to-\infty,y\to-\infty\)
• Graph 4:
• Intercepts: \(y\)-intercept \((0,3)\), \(x\)-intercept \((3,0)\)
• Slope: \(-1\)
• Positive (\(y > 0\)): \(x<3\)
• Negative (\(y < 0\)): \(x>3\)
• Domain: \(D:\{x|x\in R\}\)
• Range: \(R:\{y|y\in R\}\)
• End - Behavior: As \(x\to+\infty,y\to-\infty\); As \(x\to-\infty,y\to+\infty\)