Dave thinks that Michael Jordan could hang in the air for at least two seconds during his famous free throw line dunk. Wally disagrees with Dave. Who do you agree with and why?
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In this target, we will be using technology to graph a quadratic function and find key features of the graph. Before moving on to contextual problems, let's familiarize ourselves with how to find pieces of information such as the x-intercepts and the vertex. If you want to see how to find these features on a TI-84 Calculator please click here.
Using Desmos to Find x-Intercepts
To find the x-intercepts of a parabola on desmos, simply type the function that generates the parabola into the entry bar and click on the points where the parabola crosses the x-axis.
Example 1: Find the \(x\)-intercepts of \( y= 2x^2 -2x -4 \).
Using Desmos to Find x-Intercepts
To find the x-intercepts of a parabola on desmos, simply type the function that generates the parabola into the entry bar and click on the points where the parabola crosses the x-axis.
Example 1: Find the \(x\)-intercepts of \( y= 2x^2 -2x -4 \).
Here we are looking where our function \( y=2x^2-2x-4 \) outputs \( 0\), hence solutions to the equation \( 0=2x^2-2x-4 \) are located at the x-intercepts of the above graph, the solutions are \( x=-1 \) and \( x=2 \) since the \(x\)-intercepts are \( (-1,0) \) and \( (2, 0) \).
Recall that a parabola can also have only one \(x\)-intercept or even no \(x\)-intercepts.
Recall that a parabola can also have only one \(x\)-intercept or even no \(x\)-intercepts.
Using Desmos to Find Extrema
Finding the maximum or minimum value of a quadratic function is just as easy, simply plot the function and click on the vertex. Every parabola has a vertex, thus every parabola has either a maximum or minimum value.
Example 2: Determine if \( f(x)= -3x^2 + 6x+1 \) has a minimum or maximum value, what is this value?
Finding the maximum or minimum value of a quadratic function is just as easy, simply plot the function and click on the vertex. Every parabola has a vertex, thus every parabola has either a maximum or minimum value.
Example 2: Determine if \( f(x)= -3x^2 + 6x+1 \) has a minimum or maximum value, what is this value?
If you click on the vertex above you can see it is located at \( 1, 4\). Since the parabola opens down, this is the highest point on the graph. Hence the maximum value of \( f(x) \) is \( 4 \).
Solving Vertical Motion Problems on Desmos
The vertical motion equation to describe the time in seconds of an object in free fall is in air is given by: \( h= -16t^2 + v_0 t + h_0 \)
where \( h \) is the time at \( t \) seconds since launch/drop, \( v_0 \) is the initial vertical velocity in \( \text{ft/sec} \)which can be thought of as the speed at which the object is propelled up or down, and \( h_0 \) is the initial height of the object in feet.
where \( h \) is the time at \( t \) seconds since launch/drop, \( v_0 \) is the initial vertical velocity in \( \text{ft/sec} \)which can be thought of as the speed at which the object is propelled up or down, and \( h_0 \) is the initial height of the object in feet.
Note this model does not work for objects that are propelled straight up or down, this type of problem would require trigonometry. If you are wondering why this model is the way it is, especially why there is a \( -16 \) coefficient on \( t^2 \), come back and revisit this set up after you have taken calculus 1.
Example 3: A tennis ball is thrown upward with initial vertical velocity of 66 ft/sec from a height of 6 feet off the ground. Determine:
(a) how long it takes for the ball to reach the ground.
(b) the maximum height of the ball and how long it takes to reach this height.
First, let us set up our model to represent this problem. The initial velocity is 66 ft/sec, so \( v_0 =66\), and the starting height of the ball is 6 feet, so \( h_0=6 \). Hence, our model is: \( h= -16t^2+6t+6 \).
Example 3: A tennis ball is thrown upward with initial vertical velocity of 66 ft/sec from a height of 6 feet off the ground. Determine:
(a) how long it takes for the ball to reach the ground.
(b) the maximum height of the ball and how long it takes to reach this height.
First, let us set up our model to represent this problem. The initial velocity is 66 ft/sec, so \( v_0 =66\), and the starting height of the ball is 6 feet, so \( h_0=6 \). Hence, our model is: \( h= -16t^2+6t+6 \).
Now, we graph this model to begin our analysis.
Note: the scaling was adjusted to fit the parameters for this problem.
(a) To find how long the ball is in the air, we want to look when the height of the ball is \( 0 \). Simply click on the \( t \) intercept above, we only want the positive one! Negative \(t\)-values don't make sense here, the tennis ball isn't traveling back in time!
We find that the intercept we care about is \( (4.214, 0 ) \). This means that about \( 4.214 \) seconds after the ball is launched, it will hit the ground.
(b) To find how the maximum height, we simply look at the vertex. By clicking the graph above you can see it is \( (2.063, 74.063) \). The \(t\)-value is the time it takes to reach this point, and the \(h\)-value is the height. Thus the maximum height of the ball is about \( 74.063 \) feet and it takes about \( 2.063 \) seconds from launch to reach this height.
Bonus: look at the \(y\)-intercept, where does it show up in our model? Why?
(a) To find how long the ball is in the air, we want to look when the height of the ball is \( 0 \). Simply click on the \( t \) intercept above, we only want the positive one! Negative \(t\)-values don't make sense here, the tennis ball isn't traveling back in time!
We find that the intercept we care about is \( (4.214, 0 ) \). This means that about \( 4.214 \) seconds after the ball is launched, it will hit the ground.
(b) To find how the maximum height, we simply look at the vertex. By clicking the graph above you can see it is \( (2.063, 74.063) \). The \(t\)-value is the time it takes to reach this point, and the \(h\)-value is the height. Thus the maximum height of the ball is about \( 74.063 \) feet and it takes about \( 2.063 \) seconds from launch to reach this height.
Bonus: look at the \(y\)-intercept, where does it show up in our model? Why?
Quick Check
A swimmer decides to dabble in some minor league diving. She decides to give it her best and dive off the high diving board, which is \(10\) feet above the surface of the water. Before jumping, she bounced on the diving board a few times to give herself a starting speed of \(30\) feet per second.
1) After how many seconds does the diver reach her maximum height?
2) How long does it take for the diver to reach the surface of the water?
Round your answers to the nearest hundredth.
Quick Check Solutions
A swimmer decides to dabble in some minor league diving. She decides to give it her best and dive off the high diving board, which is \(10\) feet above the surface of the water. Before jumping, she bounced on the diving board a few times to give herself a starting speed of \(30\) feet per second.
1) After how many seconds does the diver reach her maximum height?
2) How long does it take for the diver to reach the surface of the water?
Round your answers to the nearest hundredth.
Quick Check Solutions