Quiz 040917
Choose the true statement(s) from below
If value(t) is the value of a certain item t years after its purchase, then
value(7.00 ) is its value 7.00 years after purchase.
value(t) = value after t years so
value(7) = value after 7 years.
If value(t) is the value of a certain item t years after its purchase, then the
change in its value between t = 7.00 and t = 15.0 is
value(15) = value after 15 years
value(7) = value after 7 years.
so
value(15) - value(7) is the change in the value between t = 7 and t = 15.
If value(t) is the value of a certain item t years after its purchase, then the
rate of change in its value with respect to time between t = 7.00 and t = 15.0
is
Rate of change of value with respect to clock time is change in value / change in clock time.
value(15) = value after 15 years
value(7) = value after 7 years.
so
value(15) - value(7) is the change in the value between t = 7 and t = 15.
Between t = 7 and t = 15 the change in clock time is 15 - 7 = 8.
So ave rate of change is
If value(t) is the value of a certain item t years after its purchase, then
the rise of a graph of value(t) vs. t between t = 7.00 and t = 15.0 is
value(15.0 ) - value(7.00 ).
value(7) is the vertical coordinate of the graph at t = 7
value(15) is the vertical coordinate of the graph at t = 15
The rise is the change in the vertical coordinate so the rise is value(15) - value(7).
If value(t) is the value of a certain item t years after its purchase, then the
run of a graph of value(t) vs. t between t = 7.00 and t = 15.0 is 15.0 - 7.00 .
Right.
To find the time t when the value is 411. solve the equation value(t) = 411. .
value(t) is the value at clock time t, so the equation
value(t) = 411
represents the condition that value = 411.
If we had the formula for valut(t) we could solve this
equation.
If you have a graph of value(t) vs. t you can find an approximate solution to
the equation value(t) = 411. by locating 411. on the vertical axis, projecting
straight across to the graph, then projecting down to the horizontal axis. The
solution will be the t obtained on the horizontal axis.
This is what we do.
If you have a graph of value(t) vs. t you can estimate the value at 15.0 years
by locating 15.0 on the horizontal axis, projecting straight up to the graph,
then projecting across to the vertical axis. Value(t) will be the number you end
up at on the vertical axis.
cn If value(t) is the value of a certain item t years after its purchase, then
its value 7.00 years after purchase is value(15.0 ).
n If value(t) is the value of a certain item t years after its purchase, then
the change in its value between t = 7.00 and t = 15.0 is 15.0 - 7.00 .
n If value(t) is the value of a certain item t years after its purchase, then
the rate of change in its value with respect to time between t = 7.00 and t =
15.0 is value(15.0 ) - value(7.00 ) .
n If value(t) is the value of a certain item t years after its purchase, then
the rise of a graph of value(t) vs. t between t = 7.00 and t = 15.0 is 15.0 -
7.00 .
n If value(t) is the value of a certain item t years after its purchase, then
the run of a graph of value(t) vs. t between t = 7.00 and t = 15.0 is value(15.0
) - value(7.00 )..
n To find the time t when the value is 411. solve the equation = value(411. ).
n If you have a graph of value(t) vs. t you can find an approximate value at
15.0 years by locating 15.0 on the vertical axis, projecting straight across to
the graph, then projecting down to the horizontal axis. The solution will be the
t obtained on the horizontal axis.
n If you have a graph of value(t) vs. t you can estimate the solution to the
equation value(t) = 411. by locating 411. on the horizontal axis, projecting
straight up to the graph, then projecting across to the vertical axis. Value(t)
will be the number you end up at on the vertical axis.
Question:
For the quadratic function y(t) = .0900 t^2 + -1.10 t + 57.0
The vertex will lie at t = -(-1.10 ) / (2 * .0900 ) = 6.11 .
The axis of symmetry will be the vertical line t = -(-1.10 ) / (2 * .0900 ) =
6.11 .
The vertex will be the point (6.11 , 53.6 ).
If the graph crosses the t axis it will be at the points where
If (-1.10 )^2 - 4 ( .0900 ) ( 57.0 )
) is positive the graph will cross the t
axis at two points, if negative the graph will not cross the t axis, and if zero
the graph will touch the t axis in exactly one point.
The y intercept of the graph of y(t) vs. t is the point (0, 57.0 ).
The vertical line t = 5.00 will pass through the graph at the point (5.00 ,
53.8 ).
The horizontal line y = 5.00 will pass through the graph at any value of t
which solves the equation .0900 t^2 + -1.10 t + 57.0 = 5.00 .
n The vertex will lie at t = (-1.10 ) / (2 * .0900 ) = 6.11 .
n The axis of symmetry will be the vertical line t = (-1.10 ) / (2 * .0900 ) =
6.11 .
n The vertex will be the point (53.6 , 6.11 ).
n If the graph crosses the t axis it will be at the points where t = ( (-1.10 )
+- sqrt( (-1.10 )^2 - 4 ( .0900 ) ( 57.0 ) ) / (2 * (-.0900 ) ) .
n If (-1.10 )^2 - 4 ( .0900 ) ( 57.0 ) ) is positive the graph will cross the t
axis at exactly points, if negative the graph will not cross the t axis, and if
zero the graph will touch the t axis in exactly two points.
n The y intercept of the graph of y(t) vs. t is the point (57.0 , 0).
n The vertical line t = 5.00 will pass through the graph at the point (5.00 ,
53.8 )
n The horizontal line y = 5.00 will pass through the graph at y = .0900 (5.00
)^2 + -1.10 *5.00 + 57.0 .
Question:
For a quadratic function y = a t^2 + b t + c
A vertical line will pass through the graph at exactly one point, no more or
no less.
There is
At least one horizontal line
which
will pass through the graph at two points.
At least one horizontal line will fail to pass through the graph.
At least one horizontal line will pass through the graph at one but not at two
points.
If the graph passes through the t axis at two points, those points are at
equal distances on either side of the axis of symmetry.
n A horizontal line will pass through the graph at exactly one point, no more or
no less.
n At least one vertical line will pass through the graph at two points.
n At least one vertical line will fail to pass through the graph.
At least one vertical line will pass through the graph at one but not at two
points.
n If the graph passes through the t axis at two points, the point to the left of
the axis of symmetry is closer to the axis than the point to the right.
n If the graph passes through the t axis at two points, the point to the right
of the axis of symmetry is closer to the axis than the point to the left.
For the quadratic function y(t) = .0200 t^2 + -1.30 t + 72.0
The rise between the t = 7.00 and t = 14.0 points is y(14.0 ) - y(7.00 ) =
57.7 - 72.0 = -14.3 .
The run between the t = 7.00 and t = 14.0 points is 14.0 - 7.00 = 7.00 .
The average rate of change of y with respect to t between t = 7.00 and t =
14.0 is -14.3 / 7.00 = -2.05 .
The graph point corresponding to t = 7.00 is (7.00 , 72.0 ).
The graph point corresponding to t = 14.0 is (14.0 , 57.7 ).
The average slope of the graph between t = 7.00 and t = 14.0 is -14.3 / 7.00 =
-2.05 .
n The run between the t = 7.00 and t = 14.0 points is y(14.0 ) - y(7.00 ) = 57.7
- 72.0 = -14.3 .
n The rise between the t = 7.00 and t = 14.0 points is 14.0 - 7.00 = 7.00 .
n The average rate of change of y with respect to t between t = 7.00 and t =
14.0 is 7.00 / -14.3 = -.490 .
n The graph point corresponding to t = 7.00 is (72.0 , 7.00 ).
n The graph point corresponding to t = 7.00 is (7.00 , 57.7 ).
n The graph point corresponding to t = 7.00 is (14.0 , 72.0 ).
n The graph point corresponding to t = 14.0 is (14.0 , 72.0 ).
9. What are the values of y(10) and y(10.3)?
10. It you calculate ( y(10.3) - y(10) ) / ( 10.3 - 10) what do you get, and what is the meaning of this result?
11. On the given graph of depth vs. clock time observations, sketch the t = 0, t = 1, t = 2, t = 3 and t = 4 points of the basic quadratic fuction y = x^2. In what ways does the graph of y = x^2 differ from the graph of y(t) = 0.0062 t^2 - 0.6497 t + 15.21?
Note: The To Be Memorized link now includes Properties of a Quadratic Function, as listed below.
1. Quadratic Functions have the form y(x) = a x^2 + b x + c.
2. The graph of a quadratic function is a parabola.
3. The parabola is symmetric about its vertical axis of symmetry.
4. The vertex is the extreme point of the parabola and lies on the axis of symmetry.
5. The zeros of the function lie at x = (-b +- sqrt(b^2 - 4 a c) ) / (2a).
6. The axis of symmetry of the function is the line x = - b / (2 a), which lies halfway between the zeros as long as the function has real zeros.
paper work