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(15.0 ) - value(7.00 ).
   

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

• ( value(15.0 ) - value(7.00 ) ) / (15.0 - 7.00 ).
   

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

• change in value / change in clock time = (value(15) - value(7) ) / (15 - 7).

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

• t = ( -(-1.10 ) +- sqrt( (-1.10 )^2 - 4 ( .0900 ) ( 57.0 ) ) / (2 * (.0900 ) ) .
   

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

• graph of y = x^2 shown; sketch graph of y = 2 x^2; on another copy graph y = -1/2 x^2; on another y = (x - 1)^2; on another y = x^2 - 1.  Move into basic families, shifts of points, etc..