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Behavior of Specific Functions:
(from an Algebra 1 perspective)
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Linear Functions:
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Linear functions are straight
lines. Form being examined is y = mx + b.
Some Observations:
1. If the slope, m, is positive,
the line slants uphill. As the slope gets larger, the
uphill slant of the line gets steeper. As the slope gets
extremely large (a very big number), the line becomes nearly
vertical. If the line is vertical, the slope is undefined
(because it has no horizontal change).
2. As the slope gets smaller (closer to
zero), the line loses steepness and starts to flatten. If
the slope is zero, the line is horizontal (flat),
3. If the slope is negative the line
slants downhill. As the slope decreases (remember -2 is > -3),
the downhill slant of the line gets steeper.
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Quadratic Functions:
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Quadratic functions are
parabolas.
Form being examined is y = ax2 .
Some Observations:
1. If the coefficient of x2
gets larger, the parabola becomes thinner (narrower), closer to
its line of symmetry.
2. If the coefficient of x2
gets smaller, the parabola becomes thicker (wider), further from
its line of symmetry.
3. If the coefficient of x2
is negative, the parabola opens downward.
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Absolute Value Functions:
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Notice that the graphs of these absolute
value functions are on or above the x-axis. Absolute
value always yields answers which are positive or zero. |
The graph of the absolute value
function takes the shape of a V.
Form being examined is y = | x |.
Some Observations:
1. As the coefficient of x gets
larger, the graph becomes thinner, closer to its line of symmetry.
2. As the coefficient of x gets
smaller, the graph becomes thicker, further from its line of
symmetry.
3. If the coefficient of x is
negative, the graph is the same as if that coefficient were
positive. It is acted upon by the absolute value property.
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Exponential Functions: |
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Notice that all of the exponential graphs pass through the point
(0,1). This occurs because values raised to the zero power
equal 1. |
Exponential functions have the
variable x as an exponent.
Form being examined is y = a x .
Some Observations:
1. As the base value (
a ) gets larger, the
graph becomes steeper faster - it appears to stretch upward more quickly.
2. As the base value (
a ) gets closer to 1,
the graph flattens. If the base were to become one, the graph
would be a
horizontal (flat) straight line (not an exponential graph).
3. If the base value (
a ) is between 0 and 1,
the graph appears to have reflected itself over the y-axis.
The graph has just turned from exponential growth to exponential
decay.
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