a Begin by graphing the function. To do that find it intercepts and vertex.
B
b Which point on the graph represents the maximum height?
A
aDomain: 0≤ x ≤ 2.03
Range: 0≤ y ≤ 305/16
B
b 305/16 feet
Practice makes perfect
a To determine the domain and range of the function, we should first graph the function. A quadratic function can be graphed by finding its intercepts and vertex. Let's first find its y-intercept by substituting t= 0.
At this step, we will use the Quadratic Formula to find the x-intercepts.
ax^2+ bx+ c=0 ⇔ x=- b± sqrt(b^2-4 a c)/2 a
Let's identify the values of a, b, and c.
16t^2-30t-5=0 ⇔ 16t^2 -30t -5=0
We see that a= 16, b= - 30, and c= - 5. Let's substitute these values into the Quadratic Formula.
We found two x-intercepts, (-0.15,0) and (2.03,0). Finally, we will find the coordinates of the vertex. The x-coordinate can be found by using the formula for the axis of symmetry, x=- b2a. In this case, a=-16 and b=30.
The vertex of the equation is the point ( 1516, 30516). Now we can graph the equation by plotting the intercepts and the vertex. Notice that the coefficient of the quadratic term is negative, a<0. Therefore, the graph will open down.
In our situation, time and height cannot be negative. Therefore, the graph must be bound by the axes.
Looking at the graph, we can say that the values between 0 and 2.03 form the domain and the values between 0 and 30516 form the range.
Domain:& 0≤ x ≤ 2.03
Range:& 0≤ y ≤ 305/16
b The maximum height the will reach is represented by the vertex of the graph in Part A.
