Graphing Quadratic Functions Introduction Lesson Assessment

Introduction Subject: Pre-CalculusSubject: Pre-Calculus Grade Level: 12 th gradeGrade Level: 12 th grade

Objectives The students will be able to calculate the vertex of any quadratic functionThe 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 functionThe 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+cThe 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 slideClick 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 slideClick 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 materialAfter 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) = 3x f (x) = ¼x – 2x 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) Vertex (minimum)

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

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

Assessment 1.List the a, b, and c values in order for the given quadratic function: f(x) = ½x -3x² f(x) = ½x -3x² ½, -3, 0 -3, ½, 1 3, ½, 2 -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 – 2x² – 5x keep going on forever? UP DOWN RIGHT LEFT

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

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

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

Think hard, you can do it! Hint: Think of the formula for finding the point (x,y) that gives you the vertex of a parabola Try again…

References Google images Math GV graphing program Graphing quadratic functions website