Finding the Zeroes

Review Standard Form of a Quadratic Equation y = Ax² + Bx + C

y = Ax² + Bx + C We used standard form to create a table of values to be able to graph a quadratic The ‘a’ value from standard form told us if our parabola was concave up or concave down From our graph, we could also find the line that split our parabola evenly in half We called this line the Axis of Symmetry

Sometimes it is not always easy or convenient to create a table of values to graph a quadratic Instead, we may want to find where our parabola is crossing the x – axis and the y – axis We call these points the intercepts This can be of much more help when graphing

Today we want to look at Where does the parabola go through the y-axis? (Y INTERCEPT) Where does the parabola go through the x-axis? (X INTERCEPT) How many times does the parabola cross the x-axis?

Finding y- intercepts To find the y-intercept of any function, we set our x-value = 0 and solve our equation for y

The y-axis Find the y-intercept of y = -x² - 2x + 3 Step 1: Set x = 0 Step 2: Solve for y

Find the y-intercept y = 3x² + x – 2 Step 1: Set x = 0 Step 2: Solve for y y = 3x² + x – 2

Find the y-intercept 9 + y = 0.8x² + 3x Step 1: Set x = 0 Step 2: Solve for y 9 + y = 0.8x² + 3x

Find the y-intercept y – 4 + x = -8x² Step 1: Set x = 0 Step 2: Solve for y y – 4 + x = -8x²

Can we notice a pattern for finding the y – intercept when the quadratic is given to us in standard form? For quadratics in the form y = ax² + bx + c, the y-intercept is always (0, c) Let’s look at one graphically

Axis of Symmetry: Direction of Opening a = Y-intercept =

The x-axis Another thing we want to know about parabolas is where they cross the x-axis The points where a parabola crosses the x-axis are called the x-intercepts. When talking about quadratics, we also call these points the zeroes, roots, or solutions

Finding x - intercepts To find x-intercepts, we use the same method as finding y-intercepts except we set y = 0 y = What do we do from here? Are there methods we know for solving this equation?

Finding x – intercepts To find x – intercepts of a quadratic equation, we want to convert it into the form we know as intercept form

Intercept Form We can convert quadratics in standard form into what we call intercept-form This form will show us where the graph crosses the x-axis We can rewrite our equation as y = a(x – s)(x – t)

How can we change Ax² + Bx + C to A(x – s)(x – t) ? We know this method better as factoring

y = a(x – s)(x – t) If we FACTOR our quadratic equation into two brackets, it will give us the values where our parabola crosses the x-axis In this intercept form, we can say our quadratic crosses the axis at x = s and x = t We call this method Finding the Zeroes, Finding the Roots, or Finding the Solutions

Steps for Solving Quadratics Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

Examples in Intercept Form Find the x – intercepts of the following quadratics y = (x – 2)(x + 3) y = (x + 4)(x – 1) y = (2x + 3)(3x – 4) Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

What are the x-intercepts of the following quadratics in standard form? Set y = 0, solve for x y = x² + 5x + 6 y = x² - 4x + 4 Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

What are the x-intercepts of the following quadratics in standard form? Set y = 0, solve for x y = 7x² + x – 8 y = 14 – x – 3x² Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

There are three possibilities A quadratic function has three possibilities for number of zeroes 1. No Zeroes 2. One Zero 3. Two Zeroes

No Zeroes In what cases will a quadratic function have no zeroes?

One Zero In what cases will a quadratic function have one zero?

Two Zeroes In what case will a quadratic function have two zeroes?

Axis of Symmetry: Zeroes: Direction of Opening Y-intercept =

Using our Intercepts to Graph If we can plot our intercepts on a graph, we can use the axis of symmetry to help us complete the rest of the graph The axis of symmetry will divide our two zeroes equally in half This will be where our maximum or minimum point passes through Let’s look at some examples

y = -2x² + 8x + 8 y = x² + 5x + 6

y = -0.5x² + x – 1.5 y = x² - 2x - 24

y = -2x² + 8x + 8 y = x² + 5x + 6 y = -0.5x² + x – 1.5 We can find the axis of symmetry by adding our two zeroes and dividing it by two. This is finding the center which is where our AOS passes through y = -2x² + 8x + 8 y = x² + 5x + 6 y = -0.5x² + x – 1.5 y = x² - 2x - 24