Graphing Quadratic Functions Introduction Lesson Assessment Introduction Subject

Graphing Quadratic Functions Introduction Lesson Assessment

Introduction • Subject: Pre-Calculus • Grade Level: 12 th grade

Objectives • The students will be able to calculate the vertex of any quadratic function • The students will be able to label the axis of symmetry of any given quadratic function • The students will be able to sketch the graphs of simple quadratic functions given in the form of f(x)=ax 2+bx+c

User Guide • Click the home button to go to the title slide • Click the forward or backward arrow button at the bottom of each slide to respectively move to the next slide or go back to the previous slide • After going through the lesson, try the assessment to see how well you understand the material

Lesson A quadratic function is written in the form f (x) = ax 2 + bx + c where a, b, and c are constants (number coefficients) Examples: f (x) = 3 x 2 + 4 f (x) = ¼x – 2 x 2 – 7 We can say that a= 3, b= 0, and c= 4 a= -2, b= ¼, and c= -7

The Basic Graph The graph of any quadratic equation has the shape of a tall arch. We call this shape a PARABOLA. A parabola has one “hump” and its two sides will extend outward forever – it never stops going.

Where does it go? - A quadratic function opens UPWARD if the “a” value is a positive number. - A quadratic function opens DOWNWARD if the “a” value is a negative number. +a -a

The Vertex of a Parabola The vertex of a parabola is the point (x, y) on the graph located at the top (maximum) or the bottom (minimum) of the arch: Vertex (maximum) (minimum)

Finding the Vertex The x-coordinate of the vertex (x, y) can be calculated as: X = -b/(2 a) The y-coordinate of the vertex can be found by solving: y = f (-b/2 a) For example, f (x) = x 2 – 4 x + 1 Since a=1, b= -4, and c=1: X = - (- 4) / 2(1) = 4/2 = 2 To find y = f (2), plug in (2) everywhere there is an x in (x) and simplify: f(2) = (2)2 – 4(2) + 1 =4– 8+1=-3 Therefore, the vertex of f(x) is the point (2, -3) f

Are you ready? You have completed the lesson on graphing quadratic functions… Think you can handle the quiz? ? Absolutely! Um, I think I need to review some more

Assessment 1. List the a, b, and c values in order for the given quadratic function: f(x) = ½x -3 x² ½, -3, 0 3, ½, 2 -3, ½, 1 -3, ½, 0

Good Job! You remembered the general formula for quadratic functions: f (x) = ax 2 + bx + c Next question…

Hmm… not quite! Hint: Think of the general formula for all quadratic functions Try again…

Here’s another question… 2. In which direction does the graph of f(x) = 3 – 2 x² – 5 x keep going on forever? LEFT UP RIGHT DOWN

Good Job! You remembered to look at the sign of the “a” value And you knew that a negative “a” means the graph opens, or goes on forever, in the downward direction Let’s try a harder one…

Nope, sorry! Hint: Do you remember what the sign of the “a” value means? Try again…

Ok, last one… 3. What are the coordinates of the vertex of this quadratic function: f(x) = x² + 4 x + 5 (2, 17) (-4, 5) (-2, 1) (-8, 37)

Good Job! You remembered that the vertex (x, y) is given by X = -b/2 a and y = f (-b/2 a) You really seem to understand the basic graph of a quadratic function. I’m impressed! References