How do you use the important points to sketch the graph of # f(x)= -x^2-3x-6#?

Answer 1

#"see explanation"#

# "the main points in sketching the graph are" #
#• "the x and y intercepts" #
#• "shape, minimum / maximum" #
#• "coordinates of the vertex" #
"to obtain the intercepts" #color(blue)#
#• "let x = 0, in the equation for y-intercept" #
#• "let y = 0, in the equation for x-intercepts" #
#y-intercept"# #x=0toy=-6larrcolor(red)"
#y=0to-x^2-3x-6=0#

Delta=b^2-4ac# "Check the discriminant"

#"here, #a=-1, #b=-3, and #c=-6#
(-3)^2-(4xx1xx-6)= #b^2-4ac=-15<0#
#Delta<0rArr" no x-intercepts, no real roots"#

Blue "maximum / minimum" #color

#• "If "a>0," the minimum value is "uuu#."

"If "a<0," then the maximum amount "nnn#

a=-1<0rArrnnn#"here"

"coordinates of vertex" #color(blue)

#x_(color(red)"vertex") = -b/(2a) = -(-3)/(-2) =-3 #
# "substitute this value into function for y" #

(color(red)"vertex")=(-3/2)^2-3(-3/2)-6=-15/4#

graph{-x^2-3x-6 [-20, 20, -10, 10]} #rArrcolor(magenta)"vertex" =(-3/2,-15/4)#
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Answer 2

To sketch the graph of ( f(x) = -x^2 - 3x - 6 ), follow these steps:

  1. Identify the vertex: The vertex of the parabola is given by ( (\frac{-b}{2a}, f(\frac{-b}{2a})) ). In this case, ( a = -1 ) and ( b = -3 ). So, ( x = \frac{-(-3)}{2(-1)} = \frac{3}{-2} = -\frac{3}{2} ). Plug this value of ( x ) into the function to find the corresponding ( y )-coordinate.

  2. Find the ( y )-intercept: Set ( x = 0 ) and solve for ( y ).

  3. Find the ( x )-intercepts: Set ( f(x) = 0 ) and solve for ( x ).

  4. Plot the vertex, ( y )-intercept, and ( x )-intercepts on the coordinate plane.

  5. Determine the direction of the parabola: Since the coefficient of ( x^2 ) is negative, the parabola opens downwards.

  6. Sketch the graph using the important points and the direction of the parabola.

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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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