How do you use the important points to sketch the graph of #y=x^2+4x+6#?

Answer 1

Put in zero, 1, -1, 2, -2 for x

Putting in x for zero give the value for y where the curve begins. y = 6

then putting in the small values for x fives a points to start the sketch of the graph

When x = 1 y = 11 When x = -1 y = 3

When x = 2 y = 18 When x = -2 y = 2

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Answer 2

The important points which you need for sketching a curve are:

  • the #y#-intercept
  • the #x#-intercept(s)
  • the vertex (turning point)
#y=x^2+4x+6# is the equation of a parabola.
To find the #y#-intercept, make #x=0#
#y=(0)^2 +(0) +6" "rarr y =6" "# The point is #(0,6)#
To find the #x#-intercept(s), make #y=0# and solve:
#x^2+4x+6=0# does not factorise.

Completing the square gives:

#x^2 +4x +4 =-6 +4#
#(x+2)^2 = -2#
#x+2 = +-sqrt(-2)#
There is no solution for #x#, so the curve does not cross the #x#-axis.
Find the axis of symmetry: #x = (-b)/(2a)#
#x = (-4)/(2xx1) = -2#
The vertex is on the line #x=-2#, find the #y#-value:
#y= (-2)^2+4(-2) +46= 4-8+6= 2# The vertex is at the point is #(-2,2)#
There is a point which is a reflection of the point #(0,6)# in the line of symmetry. #(-4,6)#
Plot these #3# points and draw a smooth curve to pass through all of them:

graph{y =x^2+4x+6 [-6.997, 3.003, 1.76, 6.76]}

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Answer 3

To sketch the graph of ( y = x^2 + 4x + 6 ), follow these steps:

  1. Identify the vertex: The vertex of the parabola represented by the equation ( y = ax^2 + bx + c ) is given by the coordinates ((h, k)), where ( h = -\frac{b}{2a} ) and ( k = f(h) ).

  2. Calculate the vertex: Substitute ( h = -\frac{b}{2a} ) into the equation to find the y-coordinate of the vertex.

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

  4. Find the x-intercepts (if any): Set ( y = 0 ) and solve the quadratic equation for ( x ).

  5. Plot the vertex, y-intercept, and x-intercepts on the coordinate plane.

  6. Determine the direction of the parabola: If ( a > 0 ), the parabola opens upwards. If ( a < 0 ), the parabola opens downwards.

  7. Sketch the graph: Use the important points and the direction of the parabola to sketch the graph smoothly through those points.

Remember to label the axis and any significant points on the graph.

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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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