class: center, middle, inverse, title-slide # Quadratic Functions ## Standard 2 ### MS-A4.2 Non-linear Relationships ### updated: 2022-01-16 --- # Learning Intentions #### Topic **Non-linear Relationships** #### Learning Goals Quadratic Functions #### Success Criteria * Construct and analyse a quadratic model to solve practical problems involving quadratic functions or expressions of the form `\(y=ax^2+bx+c\)`, for example braking distance against speed * recognise the shape of a parabola and that it always has a turning point and an axis of symmetry * graph a quadratic function with and without technology #### Activities/Tasks Refer to Edmodo --- # Starter question There are currently 1000 possums in a national park. The population is expected to fall by 7% each year. a) Write an exponential function for the expected possum population `\(P\)` after `\(n\)` years. b) Find the expected population after 1 year, 5 years and 10 years. c) Sketch the graph of `\(P\)` against `\(n\)` d) Use your graph to estimate the population after 3 years. --- # Quadratic Functions -- Quadratic functions are functions where the highest power on an `\(x\)` term is 2. -- i.e. `$$y=ax^2+bx+c$$` Where `\(a\)`, `\(b\)` and `\(c\)` are constants -- For example, `$$y=3x^2-4$$` --- # `\(y=x^2\)` <div id="y=x^2"></div> --- # Observations -- * There is a vertex (or turning point) at `\((0,0)\)` -- * This is the minimum value -- * The graph is symmetrical on each side of the turning point. --- # `\(y=ax^2\)` <div id="y=ax^2"></div> --- # Observations * There is a vertex (or turning point) at `\((0,0)\)` * This is the minimum value * The graph is symmetrical on each side of the turning point. --- # Observations * There is a vertex (or turning point) at `\((0,0)\)` * This is the maximum or minimum value * The graph is symmetrical on each side of the turning point. -- * `\(a\)` changes how quickly the graph increases and decreases -- * When `\(a\)` is positive it makes a 🙂 -- * When `\(a\)` is negative it makes a 🙁 -- * `\(y=ax^2\)` and `\(y=-ax^2\)` are reflections in the `\(x\)` axis --- # `\(y=ax^2+c\)` <div id="y=ax^2+c"></div> --- # Observations * There is a vertex (or turning point) at `\((0,0)\)` * This is the maximum or minimum value * The graph is symmetrical on each side of the turning point * `\(a\)` changes how quickly the graph increases and decreases * When `\(a\)` is positive it makes a 🙂 * When `\(a\)` is negative it makes a 🙁 * `\(y=ax^2\)` and `\(y=-ax^2\)` are reflections in the `\(x\)` axis -- * Adding or subtracting a number does _not_ change the shape. It moves it up or down. -- There can be 0, 1 or 2 `\(x\)`-intercepts, and there is 1 `\(y\)`-intercept. --- # `\(y=ax^2+bx+c\)` <div id="y=ax^2+bx+c"></div> --- # Sketching quadratic graphs -- 1. Construct a table of values -- 2. Draw a number plane -- 3. Plot at least 4 points -- 4. Draw a curve that goes through the points --- # Example 1 Draw the graph of `\(y=x^2+1\)` --- # `\(y=x^2+1\)` <div id="y=x^2+1"></div> --- # Example 2 Draw the graph of `\(y=x^2-4x+3\)`. Use the graph the find: a) turning point b) axis of symmetry c) `\(y\)`-intercept d) `\(x\)`-intercepts e) minimum or maximum value --- # `\(y=x^2-4x+3\)` <div id="y=x^2-4x+3"></div> --- # Check in question Draw the graph of `\(y=x^2-4x+3\)` Use the graph the find: a) turning point b) axis of symmetry c) `\(y\)`-intercept d) `\(x\)`-intercepts e) minimum or maximum value