How do you graph the function #y=1/3x^2+1/2x# and identify the domain and range?

Answer 1

See graph below.
Domain: #(-oo,+oo)# Range: #(-3/16, +oo)#

#y=1/3x^2+1/2x#
A quadratic function with the form #ax^2+bx+c#, where #a=1/3#, #b=1/2#, and #c=0#, is represented by #y#.
As a result, we are certain that #y#'s graph will resemble a parabola.
Additionally, since #a>0#, we can infer that #y# will have a unique minimum value at the location of the symmetry axis, #x=(-b)/(2a)#.
y_min = y(-3/4)# #:.
#= 1/2(-3/4)+ 1/3(-3/4)^2#
3/16-3/8 = -3/16# #=
The #x-#intercepts where #y=0# can then be found.
#-> 1/3x^2 + 1/2x = 0#
The #x-#intercepts are x=0 or x=-3/2# when #x(1/3x+1/2)=0.
Observe that there is only one #y-# intercept for the #y# at #(0,0)#.
These crucial points allow us to draw the #y# graph that is shown below.

graph{1/3x^2+1/2x [-1.528, 1.552, 3.08, -1.508]}

In RR#, #y# is defined #forall x.
Consequently, #y#'s domain is #(-oo,+oo)#.
There is no finite upper bound for #y#, and its minimum value is #-3/16#.
Consequently, #y#'s range is #(-3/16, +oo)#.
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Answer 2

To graph the function ( y = \frac{1}{3}x^2 + \frac{1}{2}x ), follow these steps:

  1. Plot points using a table of values or use a graphing calculator.
  2. Sketch the parabolic curve connecting the points.
  3. Identify the domain as all real numbers (( (-\infty, +\infty) )).
  4. Determine the range by analyzing the vertex of the parabola and whether it opens upward or downward.

The domain is ( (-\infty, +\infty) ) and the range depends on whether the parabola opens upward or downward. If it opens upward, the range is ( [y_{\text{min}}, +\infty) ), where ( y_{\text{min}} ) is the minimum y-value. If it opens downward, the range is ( (-\infty, y_{\text{max}}] ), where ( y_{\text{max}} ) is the maximum y-value.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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