1. The vertex form of a quadratic equation is
f(x) = a(x - h)² + k
where (h, k) is the parabola formed by the equation.
2. The value of a affects the shape of the parabola. Three concrete ways showing this are:
i. When a is negative ( a < 0), then the parabola opens downward
ii. When a is positive ( a > 0), the the parabola opens upward
ii. Lastly, when a reduces, the parabola shrinks. And when a increases, the parabola expands as well.
3. Since h directly affects the value of x, then it means that when h is increased by one unit, the parabola moves to the left by one unit. Similarly, if h decreases by 1 unit, it shifts to the right by one unit. Some textbooks call this as the parabola's horizontal shift.
4. The value of k directly affects the movement of the parabola across the y-axis. That means, if k is increased, the parabola goes up. And when k decreases, the graph goes down as well.
5. We have f(x) = 1(x)² as the original function with (h, k) = (0, 0). If we reflect it, across the x-axis, that means we negate the value across.
So, we now have a new function, g(x),
g(x) = -(x)².
Based from the discussion regarding translations, if we move f(x) 5 units to the left, that means we are to increase the value of h by 5. So now, g(x) becomes
g(x) = -(x - 5)²
Applying the same concept, if we shift the graph 1 unit below, we decrease the value of k by 1. So we now have a final function of
g(x) = -(x - 5)² - 1
6. Using the same initial function with 5, we have f(x) = 1(x)² with (h, k) = (0, 0). Now, since f(x) is to be compressed by 3, g(x) becomes
g(x) = 1/3(x)²
Translating 4 units to the right means decreasing the value of h by 4 and translating 2 units upwards means increasing the value of k by 2. Thus, we have
g(x) = 1/3[x - (-4)]² + 2
Simplifying this, we'll have the new function as
g(x) = 1/3(x + 4)² + 2
