Quadratic Transformations C11T7

1. Write the vertex form of a quadratic equation.

2. Name the 3 ways that a parabola changes with different types of "a" values.

3. What does changing the "h" variable do to the graph of a quadratic?

If "h" is positive how does the parabola move? If "h" is negative?

4. What does changing the "k" variable do to the graph of a quadratic?

If "k" is positive how does the parabola move? If "k" is negative?

5. Use the description to write the quadratic function in vertex form: The parent function f (x) = x2 is reflected across the x axis and translated 5 units left and 1 unit down to create g.

g (x) =

6. The parent function f (x) = x2 is compressed by a factor of 3 and translated 4 units right and 2 units up to create g (x). Write the new function in vertex form.

g (x) =

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), 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