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Graphing Quadratics: p utting it all t ogether

Learn how to sketch graphs of quadratics using key features and solve quadratics using the quadratic formula. This guide provides step-by-step strategies and techniques.

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Graphing Quadratics: p utting it all t ogether

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  1. Graphing Quadratics: putting it all together

  2. Focus 10 Learning Goal – (HS.A-REI.B.4, HS.F-IF.B.4, HS.F-IF.C.7, HS.F-IF.C.8)= Students will sketch graphs of quadratics using key features and solve quadratics using the quadratic formula.

  3. Strategies & steps to graphing a quadratic.y = 3x2 + 2x + 4 • Determine the vertex. Use x = -b/2a • x = -2/2(3) • x = -1/3 • y = 3(-1/3)2 + 2(-1/3) + 4 • y = 11/3 or 3 2/3 • The vertex is (-1/3, 3 2/3) • Use the discriminant to determine the number of solutions. • (2)2 – 4(3)(4) • 4 – 48 • -44 • No real solutions. This means the graph will NOT cross the x-axis. • Factoring or using the quadratic formula will not help you graph this quadratic. • You need to use an “x” point left and right of -1/3in order to graph this.

  4. Strategies & steps to graphing a quadratic.y = 3x2 + 2x + 4 • Point left of and right of -1/3: (-2, _____) and (1, _____) • y = 3(-2)2 + 2(-2) + 4 • y = 12 – 4 + 4 • y = 12 • (-2, 12) • y = 3(1)2 + 2(1) + 4 • y = 3 + 2 + 4 • y = 9 • (1, 9) • Graph the vertex and the two points.

  5. Strategies & steps to graphing a quadratic.y = x2 + 8x + 16 • Use the discriminant to determine the number of solutions. • (8)2 – 4(1)(16) • 64 – 64 • 0 • One solution. This means the graph will intercept the x-axis at the VERTEX. • Factoring or using the quadratic formula will not help you graph this quadratic. (Your answers will be -4 & -4). • You need to use an “x” point left and right of -4 in order to graph this. • Determine the vertex. Use x = -b/2a • x = -8/2(1) • x = -4 • y = (-4)2 + 8(-4) + 16 • y = 0 • The vertex is (-4, 0)

  6. Strategies & steps to graphing a quadratic.y = x2 + 8x + 16 • Point left of and right of -4: (-5, _____) and (-3, _____) • y = (-5)2 + 8(-5) + 16 • y = 25 – 40 + 16 • y = 1 • (-5, 1) • y = (-3)2 + 8(-3) + 16 • y = 9 - 24 + 16 • y = 1 • (-3, 1) • Graph the vertex and the two points.

  7. Strategies & steps to graphing a quadratic.y = x2 + 6x + 5 • Use the discriminant to determine the number of solutions. • (6)2 – 4(1)(5) • 36 – 20 • 16 • Two solutions. This means the graph will intercept the x-axis two times. • Factoring or using the quadratic formula will HELP you. • Which one do you want to use? • Determine the vertex. Use x = -b/2a • x = -6/2(1) • x = -3 • y = (-3)2 + 6(-3) + 5 • y = -4 • The vertex is (-3, -4)

  8. Strategies & steps to graphing a quadratic.y = x2 + 6x + 5 • Factoring will be the quickest way to find the x-intercepts. • (x +5)(x + 1) = 0 • x + 5 = 0 • x = -5 • x + 1 = 0 • x = -1 • The x-intercepts are (-5, 0) & (-1, 0). • Graph the vertex and the x-intercepts.

  9. Strategies & steps to graphing a quadratic.y = -2x2 + 6x + 1 • Use the discriminant to determine the number of solutions. • (6)2 – 4(-2)(1) • 36 + 8 • 44 • Two solutions. This means the graph will intercept the x-axis two times. • Because 44 is not a perfect square, the x-intercepts will be irrational. • Use the quadratic formula to find the x-intercepts. • Determine the vertex. Use x = -b/2a • x = -6/2(-2) • x = 3/2 • y = -2(3/2)2 + 6(3/2) + 1 • y = -2(9/4) + 9 + 1 • The vertex is (1½ , 5 ½ )

  10. Strategies & steps to graphing a quadratic.y = -2x2 + 6x + 1 • Substitute into the quadratic formula and solve to find your x-intercepts. • Divide 6, 2 and 4 by 2 (the GCF).

  11. Strategies & steps to graphing a quadratic.y = -2x2 + 6x + 1 • Estimate the value of the irrational x-intercepts. • Graph the vertex and the x-intercepts.

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