Question

Given the following graph of a parabola, find its equation, in the form
𝑦 = (4𝑥 − 3)(4𝑥 + 3)

Answer 1

First, we need to find the vertex of the parabola. The vertex is the lowest point on the curve, also known as the minimum point, if it opens upwards. In this case, we can see that the vertex is located at (-0.375, -3.25).

Next, we need to determine whether the parabola opens upwards or downwards. We can see from the graph that the parabola opens upwards, because its vertex is a minimum point.

Now, we can use the vertex form of a parabolic equation to write the equation of the parabola:

𝑦 = a(x - h)^2 + k

where (h, k) is the vertex of the parabola. Substituting the values we found earlier, we get:

𝑦 = a(x + 0.375)^2 - 3.25

To find the value of "a", we can use one of the points on the parabola. Let's use the point (1.5, 6) on the curve. Substituting these values into the equation, we get:

6 = a(1.5 + 0.375)^2 - 3.25

9.25 = a(1.875)^2

a = 9.25 / (1.875)^2

a = 2

Substituting this value of "a" into the equation, we get:

𝑦 = 2(x + 0.375)^2 - 3.25

Expanding the equation and simplifying, we get:

𝑦 = 2x^2 + 1.5x - 2.5625

But we wanted the equation in the form 𝑦 = (4𝑥 − 3)(4𝑥 + 3), so let's try to factor the equation. We can start by factoring out the coefficient of x^2, which is 2:

𝑦 = 2(x^2 + 0.75x - 1.28125)

Now, we need to find two numbers that multiply to -1.28125 and add up to 0.75. These numbers are -1.5 and 0.53125, so we can write:

𝑦 = 2(x - 0.75)(x + 0.53125)

Expanding this equation, we get:

𝑦 = 2x^2 + 1.5x - 2.5625

which is the same equation we found earlier. Therefore, the equation of the parabola is:

𝑦 = 2(x - 0.75)(x + 0.53125)

or

𝑦 = (4𝑥 − 3)(4𝑥 + 3) / 8