Write the equation of a parabola, in standard form, that goes through these points:
(0, 3) (1, 4) (-1, -6)
ax^2 + bx + c = y
a * 0^2+b*0+c = 3
a*1^2+b+1+c = 4
a*(-1)^2+b(-1)+c = -6
c = 3
a + b + c = 4
a – b + c = -6
a + b + 3 = 4
a – b + 3 = -6
Graph the parabola above. Indicate the vertex and axis of symmetry.
(the graph has a max and min number of -20 to 20, both x and y)
why stop there? subtract the equations and you get
2b = 10
b = 5
so, a = -4
so, the equations is
-4x^2+5x+3 = 0
as always, the axis of symmetry is at x = -b/2a, so it is
x = 5/8
and the vertex is at (5/8,73/16)
you can see this by writing it as
y = -4x^2+5x+3
= -4(x^2 - 5/4 x) + 3
= -4(x^2 - 5/4 x + 25/64) + 3 + 4*25/64
= -4(x - 5/8)^2 + 73/16
To find the equation of a parabola in standard form that goes through the given points (0, 3), (1, 4), and (-1, -6), we will use the system of equations method.
Step 1: Set up the equations using the given points.
Using the equation for a parabola in standard form (ax^2 + bx + c = y), we substitute the x and y values of the given points into the equation.
For the point (0, 3):
a * 0^2 + b * 0 + c = 3
c = 3
For the point (1, 4):
a * 1^2 + b * 1 + c = 4
a + b + c = 4
For the point (-1, -6):
a * (-1)^2 + b * (-1) + c = -6
a - b + c = -6
Step 2: Solve the system of equations.
Using the values we found for c in Step 1, substitute them into the other equations.
For the equation a + b + c = 4:
a + b + 3 = 4
a + b = 1 (Equation 1)
For the equation a - b + c = -6:
a - b + 3 = -6
a - b = -9 (Equation 2)
Solve the system of equations (Equation 1 and Equation 2) to find the values of a and b.
Adding Equation 1 and Equation 2, we get:
2a = -8
a = -4
Substitute the value of a into Equation 1:
-4 + b = 1
b = 5
We have found the values of a = -4 and b = 5.
Step 3: Write the equation of the parabola.
Substitute the values of a, b, and c into the standard form equation of a parabola.
The equation of the parabola is:
y = -4x^2 + 5x + 3
Step 4: Graph the parabola.
To graph the parabola, plot the points (0, 3), (1, 4), and (-1, -6) on a coordinate plane. Connect the points smoothly following the path of the parabola given by the equation.
The vertex of the parabola can be found by using the formula x = -b /(2a). Substituting the values of a = -4 and b = 5 into the formula, we get:
x = -(5) / (2 * (-4))
x = -5 / -8
x = 5/8
The axis of symmetry is a vertical line passing through the vertex. In this case, the equation of the axis of symmetry is x = 5/8.
The graph of the parabola should have a maximum or minimum point. Since we have not mentioned whether the parabola opens upwards or downwards, we cannot determine the exact position of this point without further information.
Note: The given range of -20 to 20 for both x and y suggests that you should scale your graph accordingly to fit within this range.