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8. 1 Graphing Quadratic Functions MA. 912. A. 7. 1 Graph quadratic equations. MA. 912. A. 7. 6 Identify the axis of symmetry, vertex, domain, range, and intercept(s) for a given parabola
Quadratic Function y = ax 2 + bx + c Quadratic Term Linear Term Constant Term 2 What is the linear term of y = 4 x – 3? 0 x 2 What is the linear term of y = x - 5 x ? -5 x What is the constant term of y = x 2 – 5 x? 0 Can the quadratic term be zero? No!
Quadratic Functions parabola The graph of a quadratic function is a: A parabola can open up or down. y Vertex 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 NOTE: if the parabola opens left or right it is not a function! x
Standard Form The standard form of a quadratic function is: y = ax 2 + bx + c y The parabola will open up when the a value is positive. a>0 The parabola will open down when the a value is negative. x a<0
Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. Axisy of Symmetry 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 passes through the vertex.
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GRAPH Y= x² +4 x +3 x
Finding the Axis of Symmetry When a quadratic function is in standard form y = ax 2 + bx + c, the equation of the Axis of symmetry is This is best read as … ‘the opposite of b divided by the quantity of 2 times a. ’ Find the Axis of symmetry for y = 3 x 2 – 18 x + 7 a = 3 b = -18 The Axis of symmetry is x=
Finding the Vertex The Axis of symmetry always goes through the Vertex Thus, the Axis of symmetry gives _______. X-coordinate of the vertex. us the ______ Find the vertex of y = -2 x 2 + 8 x - 3 STEP 1: Find the Axis of symmetry a = -2 b=8 The xcoordinate of the vertex is 2
Finding the Vertex Find the vertex of y = -2 x 2 + 8 x - 3 STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex. The vertex is (2 , 5)
Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. STEP 1: Find the Axis of symmetry using: 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.
Graphing a Quadratic Function y STEP 1: Find the Axis of symmetry STEP 2: Find the vertex Substitute in x = 1 to find the y – value of the vertex. x
Graphing a Quadratic Function STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. x y 2 – 1 3 5 y x
Y-intercept of a Quadratic Function Y-axis y The y-intercept of a Quadratic function can Be found when x = 0. x The constant term is always the y- intercept
Solving a Quadratic The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation. The number of real solutions is at most two. No solutions One solution X=3 Two solutions X= -2 or X = 2
Identifying Solutions Find the solutions of 2 x - x 2 = 0 The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2 x – x 2 The xintercepts(or Zero’s) of f(x)= 2 x – x 2 are the solutions to 2 x - x 2 = 0 X = 0 or X = 2