on a graph of 2x^2 -7+5 use equations to find the vertex point ,roots and asis of symmetry
If you have covered differentiation (calculus), equate f'(x) = 0 to find the value of x through which passes the axis of symmetry. Substitute into f(x) to get the value of y. The solutions of f(x)=0 can be found by either factoring or the quadratic formula.
Without using calculus, a quadratic can be transformed into its canonical form:
f(x) = a(x-h)² + k
by completing the square.
x=h is the axis of symmetry which passes through the vertex.
The vertex is the point (h, f(h))
And the roots are
h ± sqrt(-k/a)
For the case in point, assuming the correct expression is 2x²-7x+5,
2x²-7x+5
=2(x² -(7/2)x) + 5
=2(x-7/4)² + 5 - 2(7/4)²
=2(x-7/4)² - 9/8
Thus a=2, h=-7/4, k=-9/8
The axis of symmetry is x=7/4
The vertex is (7/4, f(7/4)=-9/8)
The roots are
7/4 + sqrt(-(-9/8)/2) = 5/2
7/4 - sqrt(-(-9/8)/2) = 1
Try to reproduce the answer and thus provide a check of the calculations.
A sketch of the function is shown at the following link:
http://i263.photobucket.com/albums/ii157/mathmate/anonymous.png
To find the vertex point, roots, and axis of symmetry for the given quadratic equation, 2x^2 - 7x + 5, we can use the standard form of a quadratic equation, which is given by:
y = ax^2 + bx + c
Comparing this with our equation, 2x^2 - 7x + 5, we can determine the values of a, b, and c:
a = 2
b = -7
c = 5
Now, let's start by finding the vertex point.
1. Vertex Point:
The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
Substituting the values, we have:
x = -(-7) / (2*2)
x = 7/4
To find the y-coordinate of the vertex, substitute this value of x into the equation:
y = 2(7/4)^2 - 7(7/4) + 5
y = 2(49/16) - 49/4 + 5
y = 98/16 - 49/4 + 5
y = 98/16 - 196/16 + 80/16
y = -18/16
y = -9/8
Therefore, the vertex point is (7/4, -9/8).
2. Axis of Symmetry:
The equation for the axis of symmetry is given by: x = -b / (2a)
Substituting the values, we have:
x = -(-7) / (2*2)
x = 7/4
So, the axis of symmetry is x = 7/4.
3. Roots (x-intercepts):
To find the roots, we need to set the equation equal to zero and solve for x. This is done by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values, we have:
x = (-(-7) ± √((-7)^2 - 4*2*5)) / (2*2)
x = (7 ± √(49 - 40)) / 4
x = (7 ± √9) / 4
x = (7 ± 3) / 4
This gives us two possibilities for the roots:
x1 = (7 + 3) / 4 = 10/4 = 5/2
x2 = (7 - 3) / 4 = 4/4 = 1
Therefore, the roots are x = 5/2 and x = 1.
To summarize:
- Vertex Point: (7/4, -9/8)
- Axis of Symmetry: x = 7/4
- Roots: x = 5/2 and x = 1