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Quadratic Functions yThe graph of a quadratic functionis a parabola. VertexA parabola can open up or down.If the parabola opens up, thelowest point is called the vertex. xIf the parabola opens down, thevertex is the highest point.NOTE: if the parabola opened Vertexleft or right it would not be afunction!
Standard Form yThe standard form of aquadratic function is a>0 y = ax2 + bx + cThe parabola will open upwhen the a value is positive. xThe parabola will open downwhen the a value is negative. a<0
Line of Symmetry LineyofParabolas have a symmetric Symmetryproperty to them.If we drew a line down themiddle of the parabola, wecould fold the parabola in half.We call this line the line of xsymmetry.Or, if we graphed one side ofthe parabola, we could “fold”(or REFLECT) it over, the lineof symmetry to graph the other The line of symmetry ALWAYSside. passes through the vertex.
Finding the Line of SymmetryWhen a quadratic function is in For example…standard form Find the line of symmetry of y = ax + bx + c, 2 y = 3x2 – 18x + 7The equation of the line ofsymmetry is Using the formula… x = −b x = 18 = 18 = 3 2a 2( 3) 6This is best read as … the opposite of b divided by the Thus, the line of symmetry is x = 3. quantity of 2 times a.
Finding the VertexWe know the line of symmetry y = –2x2 + 8x –3always goes through the vertex. STEP 1: Find the line of symmetryThus, the line of symmetrygives us the x – coordinate of x = −b = −8 = −8 = 2 2a 2(−2) −4the 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) –3vertex, 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)
A Quadratic Function in Standard FormThe standard form of a quadratic There are 3 steps to graphing afunction is given by parabola in standard form. y = ax2 + bx + c USE the A TABLE MAKE equationof Plug in the lineSTEP 1: Find the line of symmetry symmetry (x – value) to b - value of the using x – values close to obtain x = – the ySTEP 2: Find the vertex the line of symmetry. 2a vertex.STEP 3: Find two other points and reflectthem across the line of symmetry. Thenconnect the five points with a smoothcurve.
A Quadratic Function in Standard FormLets Graph ONE! Try … y y = 2x2 – 4x – 1STEP 1: Find the line ofsymmetry -b 4 x= = =1 x 2a 2( 2) Thus the line of symmetry is x = 1
A Quadratic Function in Standard FormLets Graph ONE! Try … y y = 2x2 – 4x – 1STEP 2: Find the vertexSince the x – value of thevertex is given by the line ofsymmetry, we need to plugin x = 1 to find the y – value xof the vertex. y = 2( 1) 2 - 4( 1) - 1 = - 3 Thus the vertex is (1 ,–3).
A Quadratic Function in Standard FormLets Graph ONE! Try … y y = 2x2 – 4x – 1STEP 3: Find two other pointsand reflect them across the lineof symmetry. Then connect thefive points with a smoothcurve. x y x 2 –1 3 5 y = 2 ( 2) 2 - 4 ( 2 ) - 1 = - 1 y = 2( 3) 2 - 4( 3) - 1 = 5