How do you graph and label the vertex and axis of symmetry of #y=3(x1)(x4)#?
Vertex
Axis of Symmetry
Given 
#y=3(x1)(x4)#
Let us rewrite it as 
#y=(3x3)(x4)#
#y=3x^23x12x+12#
#y=3x^215x+12#
Vertex 
#x=(b)/(2a)=((15))/(2 xx 3)=15/6=5/2#
At
#y=3(2.5)^215(2.5)+12#
#y=18.7537.5+12=6.75#
Vertex
Axis of Symmetry
Take a few values on either side of
Find the corresponding y value. Tabulate them and plot them.
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To graph and label the vertex and axis of symmetry of the quadratic function y = 3(x  1)(x  4):

First, find the vertex using the formula: Vertex = (h, k), where h = b/ (2a) and k = f(h).
In the equation y = 3(x  1)(x  4), a = 3, b = 15, and c = 12.
Substitute these values into the formula:
h = (15) / (2 * 3) = 15 / 6 = 5/2
k = 3(5/2  1)(5/2  4) = 3(1/2)(3/2) = 9/4
So, the vertex is at (5/2, 9/4).

Next, find the axis of symmetry, which is the vertical line passing through the vertex. The equation of the axis of symmetry is x = h.
In this case, x = 5/2.

Now, plot the vertex (5/2, 9/4) on the coordinate plane.

Determine two more points on the graph. Choose values of x and plug them into the equation y = 3(x  1)(x  4) to find the corresponding yvalues. A good practice is to choose xvalues symmetrically around the axis of symmetry.
For example, let's choose x = 0 and x = 3:
For x = 0: y = 3(0  1)(0  4) = 3(1)(4) = 12
For x = 3: y = 3(3  1)(3  4) = 3(2)(1) = 6

Plot these points on the coordinate plane.

Draw a smooth curve passing through these points. This curve represents the graph of the quadratic function y = 3(x  1)(x  4).

Finally, label the vertex as (5/2, 9/4) and draw a dashed line representing the axis of symmetry, which is x = 5/2.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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