Graphing quadratic equations

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  • Ask students “Why is ‘a’ not allowed to be zero? Would the function still be quadratic?
  • Let students know that in Algebra I we concentrate only on parabolas that are functions; In Algebra II, they will study parabolas that open left or right.
  • Remind students that if ‘a’ = 0 you would not have a quadratic function.
  • Discuss with the students that the line of symmetry of a quadratic function (parabola that opens up or down) is always a vertical line, therefore has the equation x =#. Ask “Does this parabola open up or down?
  • Graphing quadratic equations

    1. 1. Solving Quadratic Equations by Graphing
    2. 2. Quadratic Equation y = ax2 + bx + c 2 is the quadratic term. ax bx is the linear term. c is the constant term. The highest exponent is two; therefore, the degree is two.
    3. 3. Identifying Terms 2-7x+1 Example f(x)=5x Quadratic term 5x2 Linear term -7x Constant term 1
    4. 4. Identifying Terms 2-3 Example f(x) = 4x 2 4x Quadratic term Linear term 0 Constant term -3
    5. 5. Identifying Terms Now you try this problem. 2 - 2x + 3 f(x) = 5x quadratic term linear term constant term 5x2 -2x 3
    6. 6. Quadratic Solutions The number of real solutions is at most two. 6 f x  = x 2 -2  x +5 6 2 4 4 -5 2 5 2 -2 5 5 -4 -2 -2 No solutions One solution Two solutions
    7. 7. Quadratic Function y = ax2 + bx + c Quadratic Term Linear Term Constant Term 2 – 3? 0x What is the linear term of y = 4x 2- 5x ? -5x What is the linear term of y = x 2 – 5x? What is the constant term of y = x 0 Can the quadratic term be zero? No!
    8. 8. Solving Equations When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts. These values are also referred to as solutions, zeros, or roots.
    9. 9. Identifying Solutions 2-4 Example f(x) = x 4 2 -5 -2 -4 Solutions are -2 and 2.
    10. 10. Identifying Solutions Now you try this problem. 2 f(x) = 2x - x 4 2 5 -2 Solutions are 0 and 2. -4
    11. 11. Quadratic Functions The graph of a quadratic function is parabola a: y A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex (minimum). If the parabola opens down, the vertex is the highest point (maximum). Vertex x Vertex NOTE: if the parabola opens left or right it is not a function!
    12. 12. Standard Form The standard form of a quadratic function is: y = ax2 + bx + c y The parabola will open up when the a value is positive. The parabola will open down when the a value is negative. a0 a>0 x a<0
    13. 13. Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. y Axis of Symmetr y We call this line the Axis of symmetry. x If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side. The Axis of symmetry ALWAYS
    14. 14. Finding the Axis of Symmetry When a quadratic function is in standard form 2 y = ax + bx + c, the equation of the Axis of symmetry is This is best read as … x b 2a ‘the opposite of b divided by the quantity of 2 times a.’ 2 Find the Axis of symmetry for y = 3x – 18x a=3 b = -18+ 7 The Axis 18 18 x  2 3  6 3 of symmetry is x = 3.
    15. 15. Finding the Vertex The Axis of symmetry always goes through the Vertex _______. Thus, the Axis of symmetry X-coordinate gives us the ____________ of the vertex. Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry x b a = -2 b=8 2a x  8 2( 2)  8 4  2 The xcoordinate of the vertex is 2
    16. 16. Finding the Vertex Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry x b  2a 8 2(  2)  8 4 2 STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex. y  2  2 2 2  8  4   16   8  16  3  5 2  3  3 The vertex is (2 , 5)
    17. 17. Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. x STEP 1: Find the Axis of symmetry using: b 2a STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry.
    18. 18. Graphing a Quadratic Function Graph : y  2 x 2  4x 1 y x 1 STEP 1: Find the Axis of symmetry x= - b 2a = 4 2 (2 ) = 1  STEP 2: Find the vertex Substitute in x = 1 to find the y – value of the vertex. 2 y = 2 (1) - 4 (1) - 1 = - 3 x Vertex : 1 ,  3 
    19. 19. Graphing Quadratic Equations The graph of a quadratic equation is a parabola. The roots or zeros are the x-intercepts. The vertex is the maximum or minimum point. All parabolas have an axis of symmetry.
    20. 20. Graphing Quadratic Equations One method of graphing uses a table with arbitrary x-values. Graph y = x2 - 4x 4 2 x y 0 1 2 3 4 0 -3 -4 -3 0 5 -2 -4 Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2
    21. 21. Graphing Quadratic Equations Try this problem y = x2 - 2x - 8. 4 x y -2 -1 1 3 4 2 5 -2 -4 Roots Vertex Axis of Symmetry

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