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Vertex and Intercept Form of Quadratic Function Standard: MM2A3c Students will Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase, and decrease, and rates of change. Vertex Form of the Quadratic Determine the vertex of the following functions: •f(x) = 2(x – 1)2 + 1 •g(x) = -(x + 3)2 + 5 •h(x) = 3(x – 2)2 – 7 Vertex & Axis of Symmetry Summary Put equation in standard form f(x) = ax2 + bx + c Determine the value “a” and “b” Determine if the graph opens up (a > 0) or down (a < 0) b Find the axis of symmetry: x 2a Find the vertex by substituting the “x” into the function and solving for “y” Determine two more points on the same side of the axis of symmetry Graph the axis of symmetry, vertex, & points Vertex Form of the Quadratic Determine the vertex of the following equations: f(x) = 2(x – 1)2 + 1 V = (1, 1) g(x) = -(x + 3)2 + 5 V = (-3, 5) h(x) = 3(x – 2)2 – 7 V = (2, -7) Compare equations and theofvertices. The x partthe is the opposite sign the Do you notice pattern? number insideathe brackets and the y part is the same as the number added or subtracted at the end. Vertex Form of the Quadratic The vertex form of the quadratic equation is of the form: y = a(x – h)2 + k, where: The vertex is located at (h, k) The axis of symmetry is x = h The “a” is the same as in the standard form The “a” is the stretch of the function The vertex is shifted right by h The vertex is shifted up by k Vertex Form of the Quadratic From y = x2 Stretch factor Vertex Shift VERTICAL amount y = a(x – 2 h) +k Vertex Shift HORIZONTAL amount 6 5 fx = x+32 a>0 Graph Opens UP Vertex is a MINIMUM 4 Parent Curve: 3 hx = x2 2 gx = -x-32+5 1 -6 -4 -2 a<0 Graph Opens DOWN Vertex is a MAXIMUM 2 -1 4 6 In Class: Do page 63 of Note Taking Guide Do first 6 problems of Henley Task Day 2 – be sure to graph the y = x2 for each graph. In Class Do page 64 of the Note Taking Guide Do Day 2 of the Henley Task, # 4a – 4e all Intercept Form of the Quadratic Function How can we determine the vertex of the following equations without putting them in standard form? V = (2, -1) •f(x) = (x – 3)(x – 1) •g(x) = 2(x + 1)(x + 4) V = (-2.5, -4.5) •h(x) = -3(x – 2)(x + 3) V = (-0.5, 18.75) •Determine the x-intercepts (zero prod rule) •Find the axis of symmetry (average) •Find “y” value of the vertex (sub into f(x)) Homework Page 65, # 1, 2, and 19 – 22 all Convert from Standard to Vertex Form Standard: MM2A3a Students will Convert between standard and vertex form. Convert from Standard to Vertex Forms We converted from Vertex form to Standard form of the quadratic function above in slide 3 by expanding the (a – h)2 term and combining like terms How can we convert from Standard form to Vertex form? Convert from Standard to Vertex Forms Look at the standard form: y = ax2 + bx + c, where a ≠ 0 And look at the Vertex form: y = a(x – h)2 + k “h” is the axis of symmetry, which is the “x” part of the coordinates of the vertex “k” is the “y” part of the vertex Convert from Standard to Vertex Forms How did we find the axis of symmetry? b This is the “h” of the vertex x 2a form How did we then find the “y” part of the vertex? Substitute the x into the original equation and solve for y. This is the “k” of the vertex form The “a” is the same for both forms Convert from Standard to Vertex Forms Convert the following functions to vertex form: f(x) = x2 + 10x – 20 y = (x + 5)2 - 45 g(x) = -3x2 – 3x + 10 y = -3(x + 0.5)2 + 10.75 h(x) = 0.5x2 – 4x – 3 y = 0.5(x – 4)2 - 11