Question 347559: which direction does the parabola for the quadratic equation y = -5x2 – 16x + 21 open?
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! y=-5x^(2)-16x+21
To create a trinomial square on the left-hand side of the equation, add a value to both sides of the equation that is equal to the square of half the coefficient of x. In this problem, add ((8)/(5))^(2) to both sides of the equation.
y=-5(x^(2)+(16x)/(5)+(64)/(25))-5(-(21)/(5))-(-5)(0+(64)/(25))
Remove the 0 from the polynomial; adding or subtracting 0 does not change the value of the expression.
y=-5(x^(2)+(16x)/(5)+(64)/(25))-5(-(21)/(5))-(-5)((64)/(25))
Factor the perfect trinomial square into (x+(8)/(5))^(2).
y=-5((x+(8)/(5))^(2))-5(-(21)/(5))-(-5)((64)/(25))
Factor the perfect trinomial square into (x+(8)/(5))^(2).
y=-5(x+(8)/(5))^(2)-5(-(21)/(5))-(-5)((64)/(25))
Multiply -5 by each term inside the parentheses.
y=-5(x+(8)/(5))^(2)+21-(-5)((64)/(25))
Multiply -5 by (64)/(25) to get -(64)/(5).
y=-5(x+(8)/(5))^(2)+21-(-(64)/(5))
Multiply -1 by each term inside the parentheses.
y=-5(x+(8)/(5))^(2)+21+(64)/(5)
To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 5. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
y=-5(x+(8)/(5))^(2)+21*(5)/(5)+(64)/(5)
Complete the multiplication to produce a denominator of 5 in each expression.
y=-5(x+(8)/(5))^(2)+(105)/(5)+(64)/(5)
Combine the numerators of all fractions that have common denominators.
y=-5(x+(8)/(5))^(2)+(105+64)/(5)
Add 64 to 105 to get 169.
y=-5(x+(8)/(5))^(2)+(169)/(5)
This is the form of a paraboloa. Use this form to determine the values used to find vertex and x-y intercepts.
y=a(x-h)^(2)+k
Use the standard form to determine the vertex and x-y intercepts.
a=-5_k=(169)/(5)_h=-(8)/(5)
The vertex of a parabola is (h,k).
Vertex: (-(8)/(5),(169)/(5))
This formula is used to find the distance from the vertex to the focus.
(1)/(4p)=a
Substitute the value of a into the formula.
(1)/(4p)=-5
Solve the equation for p.
p=-(1)/(20)
Add p to the vertex to find the focus. If the parabola points up or down add p to the y-coordinate of the vertex, if it points left or right add it to the x-coordinate.
Focus=(-(8)/(5),(169)/(5)-(1)/(20))
Find the focus.
Focus=(-(8)/(5),(135)/(4))
A parabola can also be defined as locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix).
y=(169)/(5)-(-(1)/(20))
Find the directrix.
Directrix: y=(677)/(20)
The axis of symmetry is the line that passes through the vertex and focus. The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.
Axis of Symmetry: x=-(8)/(5)
These values represent the important values for graphing and analyzing a parabola.
Vertex: (-(8)/(5),(169)/(5))_Focus: (-(8)/(5),(135)/(4))_Directrix: y=(677)/(20)_Axis of Symmetry: x=-(8)/(5)
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