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- 1. Graphing Quadratic Functions y = ax2 + bx + c
- 2. All the slides in this presentation are timed. You do not need to click the mouse or press any keys on the keyboard for the presentation on each slide to continue. However, in order to make sure the presentation does not go too quickly, you will need to click the mouse or press a key on the keyboard to advance to the next slide. You will know when the slide is finished when you see a small icon in the bottom left corner of the slide. Click the mouse button to advance the slide when you see this icon.
- 3. Quadratic Functions y The graph of a quadratic function is a parabola. Vertex A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. x If the parabola opens down, the vertex is the highest point. NOTE: if the parabola opened Vertex left or right it would not be a function!
- 4. Standard Form y The standard form of a quadratic function is a>0 y = ax2 + bx + c The parabola will open up when the a value is positive. x The parabola will open down when the a value is negative. a<0
- 5. Line of Symmetry Lineyof Parabolas have a symmetric Symmetry property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of x symmetry. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other The line of symmetry ALWAYS side. passes through the vertex.
- 6. Finding the Line of Symmetry When a quadratic function is in For example… standard form Find the line of symmetry of y = ax + bx + c, 2 y = 3x2 – 18x + 7 The equation of the line of symmetry is Using the formula… x = −b x = 18 = 18 = 3 2a 2( 3) 6 This is best read as … the opposite of b divided by the Thus, the line of symmetry is x = 3. quantity of 2 times a.
- 7. Finding the Vertex We know the line of symmetry y = –2x2 + 8x –3 always goes through the vertex. STEP 1: Find the line of symmetry Thus, the line of symmetry gives us the x – coordinate of x = −b = −8 = −8 = 2 2a 2(−2) −4 the vertex. STEP 2: Plug the x – value into the original equation to find the y value. To find the y – coordinate of the y = –2(2)2 + 8(2) –3 vertex, we need to plug the x – value into the original equation. y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y=5 Therefore, the vertex is (2 , 5)
- 8. A Quadratic Function in Standard Form The standard form of a quadratic There are 3 steps to graphing a function is given by parabola in standard form. y = ax2 + bx + c USE the A TABLE MAKE equationof Plug in the line STEP 1: Find the line of symmetry symmetry (x – value) to b - value of the using x – values close to obtain x = – the y STEP 2: Find the vertex the line of symmetry. 2a vertex. STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
- 9. A Quadratic Function in Standard Form Let's Graph ONE! Try … y y = 2x2 – 4x – 1 STEP 1: Find the line of symmetry -b 4 x= = =1 x 2a 2( 2) Thus the line of symmetry is x = 1
- 10. A Quadratic Function in Standard Form Let's Graph ONE! Try … y y = 2x2 – 4x – 1 STEP 2: Find the vertex Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value x of the vertex. y = 2( 1) 2 - 4( 1) - 1 = - 3 Thus the vertex is (1 ,–3).
- 11. A Quadratic Function in Standard Form Let's Graph ONE! Try … y y = 2x2 – 4x – 1 STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. x y x 2 –1 3 5 y = 2 ( 2) 2 - 4 ( 2 ) - 1 = - 1 y = 2( 3) 2 - 4( 3) - 1 = 5