QUADRATICS

The standard form of a quadratic equation is y = ax² + bx + c

What is a quadratic? A more familiar word for quadratic is parabola This is a function that we can graph We can tell if a function is a quadratic if it is a polynomial equation of degree 2

Quadratic or not? y = 1 - x² y = 2(x – 3)(x + 4) x³ - y = 4

y = Ax² + Bx + C Write the A, B, and C values for the functions that were quadratics y = 1 - x² y = 2(x – 3)(x + 4) y + x² = 3x – 1 (x – 3)² = y + 2

The most basic form of a quadratic is y = x² In this case, A = 1, B = 0, and C = 0 For all quadratics, A ≠ 0 Why? What would this look like if we graphed it?

Table of Values One method we have for graphing parabolas is by using a table of values This allows us to plot points on a graph and estimate or sketch what we think the graph will look like

Plotting Points for y = x² The most basic way for graphing a quadratic is to make a table of values x y -2 -1 1 2

Plotting Points for y = x²

Plotting Points for y = x² + 1 A = B = C = x y

y = x² + 1

Plotting Points for y = x² -2x + 3 A = B = C = x y

y = x² -2x + 3

Plotting Points for y = -2x² + x – 1 A = B = C = x y

y = -2x² + x – 1

Plotting Points for y = 5x² + 2x – 6 A = B = C = x y

y = 5x² + 2x – 6

Plotting Points for y = 0.5x² - 4x A = B = C = x y

y = 0.5x² - 4x

Comparing to y = x² In the previous examples, our ‘a’ value can tell us if our parabola opens up or down If ‘a’ is positive, our parabola opens UP or is called concave up If ‘a’ is negative, our parabola opens DOWN or is called concave down

If our parabola is concave up This means our A value is positive We would say our parabola has a minimum value at its lowest point

If our parabola is concave down This means our A value is negative We would say our parabola has a maximum value at its highest point

Example y + x² = 3 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ x y

Example 3x² - 2x = 3 + y ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ x y

Example x² = y – 1 + x ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ x y

Example y = -0.5x² + 2x – 1 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ x

Example y + -2x² = x ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ x y

How else can we describe our parabola? So far from the graph, we have talked about our parabola opening up or down When describing parabolas from a graph, we can also talk about the axis of symmetry This tells us the equation of the vertical line that divides our parabola evenly in half Or more mathematically, it is the vertical line which divides the parabola so that one half is perfectly reflected or mirrored onto the other half Let’s look at our previous examples and find axis of symmetry

Example y = 2x² - x + 3 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ Axis of Symmetry is ______

Example y = -3x² + 2x – 1 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ Axis of Symmetry is ______

Example y = -2x² - 3x + 1 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ Axis of Symmetry is ______