Given the following graph of a parabola, find its equation, in the form
𝑦 = (4𝑥 − 3)(4𝑥 + 3)
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
To find the equation of a parabola given its graph, we need to identify the key features of the parabola. From the given graph, we can see that the parabola opens downwards and its vertex is at the point (-0.75, 0).
The vertex form of a parabola equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
By analyzing the given graph, we can determine that the vertex is at (-0.75, 0). So, h = -0.75 and k = 0.
Now, we need to find the value of a. To do so, we can use another point on the curve. From the graph, we can see that the x-intercepts are approximately -0.625 and 0.625.
Since the parabola opens downwards, the equation can be written as:
y = a(x + 0.75)^2
To find the value of a, we substitute the coordinates of either x-intercept into the equation:
0 = a(-0.625 + 0.75)^2
Let's calculate this:
0 = a(0.125)^2
0 = a(0.015625)
a = 0
Therefore, the equation of the given parabola is:
y = 0(x + 0.75)^2
y = 0
In other words, the equation simplifies to y = 0.