Quadratics

Quadratic Equations a quadratic equation is an equation of degree 2, meaning that the highest exponent of this function is 2.

Key Features of a Quadratic Function Roots (aka zeros, solutions, x-intercepts): Where the graph crosses the x-axis Line of symmetry (aka mirror line): a line through the center of the graph that splits the graph into 2 reflected images

Key Features of a Quadratic Function Vertex (aka the thing that no one wants to find algebraically): the point where the graph changes from increasing to decreasing or decreasing to increasing Max/min (aka the vertex again): the highest (max) or lowest (min) point on the graph that is not infinity

3 different forms of the quadratic equation Standard form (the least useful) –Good for quadratic formula and not much else Y-intercepta Coefficient Shows whether opens up or down and vertical dilation (stretch)

3 different forms of the quadratic equation Vertex form (the one you usually have to convert standard form to) –Best for graphing, tells you the vertex Same a as standard form H and k will be numbers, the vertex will be the coordinate (h, k) Note the (–) in front of h

3 different forms of the quadratic equation Factored form (the one you have been doing since before algebra 1) –Best for solving quadratic equations which is really finding the x-intercepts of the graph Each () is set =0 and solved individually

Example Convert to standard form, then factored form, then graph the function stating all the key features

Finding the axis of symmetry from standard form Use the formula How can we use this to find the vertex? Can we use this to convert to vertex form?

Solving for Quadratic equations aka: find the roots, solutions, zeros, x-intercepts Multiple methods depending on which form of the equations you have to work with The roots are the x-intercepts so they are located on the x-axis X-axis is when y=0 so f(x)=0

Factored Form Review Uses the zero product property Set each binomial =0 Solve each problem seperately

Quadratic Formula What does this mean on the graph? What does this part mean then? X coordinate of the vertex Horizontal distance from the vertex Also the axis of symmetry

Standard Form Option one: convert to factored form and solve Option two: Quadratic formula

Vertex Form Solve like absolute value except… Instead of making 2 equations, square root both sides Is the answer + or -?

Examples: Find the roots of each function and Graph