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Quadratic Equations

Quadratic Equations. Objectives 1. Find the vertex and standard equation of a parabola 2. Sketch the graph of a parabola 3. Solve applied problem involving maximum or minimum. Example 1. Sketch the graph of f if f( x ) = x 2.

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Quadratic Equations

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  1. Quadratic Equations Objectives 1. Find the vertex and standard equation of a parabola 2. Sketch the graph of a parabola 3. Solve applied problem involving maximum or minimum

  2. Example 1. Sketch the graph of f if • f( x ) = x2 Definition of a quadratic Function: A function f is a quadratic function if f(x) = ax2 + bx + c Where a, b and c are constants with a ≠ 0 Note: b = c = 0 Solution: Table of variation

  3. Note: Only b = 0 Solution: It is enough to shift the previous graph by 4 units upward as we can see below

  4. Method 2. ( Zalzali’s Method ) Step 1. Let h = Step 2. Let k = Step 3. Insert the values of hand k in f ( x ) = a( x – h )2 + k Expressing a Quadratic Equation asf(x) = a ( x – h )2 + k Method 1. By completing the square of the right hand side of the quadratic equation and it will be explained in class.

  5. Example 3.Write the following quadratic function in the standard form f(x) = a ( x –h )2 + k Solution: Then f(x) = 3 ( x + 4 )2 +2 Class Work 1 Write the following quadratic function f(x) = -x2 -2x +8 in the standard form f(x) = a ( x –h )2 + k Answer: f(x) = - ( x + 1 )2 +9

  6. Standard Equation of a parabola with vertical Axis The graph of the equation f(x) = a ( x –h )2 + k For a ≠ 0, the graph of f is aparabolawith vertex V(h, k ) and it has a vertical axis x = h. The parabola opens upward if a > 0 and The parabola opens downward if a < 0. Notes about Vertex V ( h , k ) Note 1. If a < 0, then the vertex V ( h, k ) is a maximum point of the parabola Note 2. If a > 0, then the vertex V ( h, k ) is a minimum point of the parabola

  7. Graphing Parabolas Strategy: 1. Find the vertex V ( h, k ) 2. Identify if the vertex is maximum or minimum 3. Find x and y intercepts if they exist 4. Plot the vertex and the intercepts 5. Plot the vertical axis ( it is also called axis of symmetry of a parabola ) 6. Connect the points of the parabola and extend the graph Example 4: Graph the parabola Solution: 1. Vertex V ( -1, 9) Check class practice f(x) = - x2 – 2x + 9 2. a = -1 < 0 ( Parabola is open downward ) and has a maximum at the vertex 3. x-intercept(s): Set y = 0, then x = - 4 and 2 Therefore, points are ( -4, 0 ) and ( 2, 0 ) y-intercept : Set x = 0 , then y = 8. Point ( 0, 8 ) x = -1

  8. Class Work 2 Graph the parabola

  9. Height =s(0) Word Problem Height of a projectile: An object is projected vertically upward from the top of a building with an initial velocity of 144 ft / sec. Its distance s( t ) = - 16t2 + 144t + 100 (a) Find its maximum distance above the ground (b) Find the height of the building. Solution:

  10. The End

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