How to find a quadratic equation just from a graph?

Question

how do I find the equation with just the turning point of (3,6), the y intercept of -3 and 2 x intercepts with 3 being directly in the middle of Them?

Answer 1

A quadratic equation represents a parabola.  The turning point of this equation is the same as the vertex of the parabola.  Since the vertex is already given, we can simply put the equation in vertex form.

 

y = a(x - h)² + k

 

where:

a is the coefficient of the x² term

(h, k) is the vertex

 

 

Plugging all this in, we get

 

y = a(x - 3)² + 6

 

 

Now we use the fact that the y-intercept is (0, -3) to find the value of a.  So we plug in x=0 and y=-3 into this equation to find a.

 

-3 = a(0 - 3)² + 6

-3 = (-3)²a + 6

-3 = 9a + 6

-9 = 9a

-1 = a

 

 

Now, we can plug in the a value we just found into the vertex form of the equation in terms of x.

 

y = -(x - 3)² + 6

 

 

Expand to put the equation in standard form.

 

y = -(x - 3)(x - 3) + 6

y = -(x² - 6x + 9) + 6

y = -x² + 6x - 9 + 6

 

y = -x² + 6x - 3      ---->  your final equation

 

 

The x-coordinate of the vertex is directly in the middle of the 2 x-intercepts, because a parabola is a symmetrical graph.  To prove this, we can find the x-intercepts by setting this equation equal to zero and solving for x.

 

0 = -x² + 6x - 3

 

 

Using the quadratic formula to solve for x:

 

x = (-b ± √(b² - 4ac)) / 2a

 

x = (-6 ± √(36 - 4(3))) / -2

 

x = (-6 ± √(36 - 12)) / -2

 

x = (-6 ± √(24)) / -2

 

x = (-6 ± 2√6) / -2

 

x₁ = (-6 - 2√6) / -2                 and                x₂ = (-6 + 2√6) / -2 

 

x₁ = 3 + √6                            and                x₂ = 3 - √6

 

 

If you take the average of x₁ and x₂, you get 3, which is the x-coordinate of the vertex.

Answer 2

Can you post the actual question or share the image link.