9-1 Exploring Quadratic Graphs Hubarth Algebra

Standard Form of a Quadratic Function A quadratic function is a function that can be written in the form 𝑦= 𝑎𝑥 2 +𝑏𝑥+𝑐, where a, b, and c are real numbers and a≠0. This form is called standard form of a quadratic function. EXAMPLE 𝑦= 5𝑥 2 𝑦= 𝑥 2 +7 𝑦= 𝑥 2 −𝑥−3 The graph of a quadratic function is a U-shaped curve called a parabola. You can fold a parabola so that the two sides match exactly. This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry. The highest or lowest point of a parabola is its vertex, which is a point on the axis of symmetry.

If a > 0 in 𝑦= 𝑎𝑥 2 +𝑏𝑥+𝑐 If a < 0 in 𝑦= 𝑎𝑥 2 +𝑏𝑥+𝑐 a is positive a is negative The vertex is the minimum point or the lowest point of the parabola. The vertex is the maximum point or highest point of the parabola.

Ex 1 Identifying a Vertex Identify the vertex of each graph. Tell whether the vertex is a minimum or a maximum. a. b. The vertex is (1, 2). The vertex is (2, –4). It is a maximum. It is a minimum.

Make a table of values and graph the quadratic function y = x2. Ex 2 Graphing 𝒚= 𝒂𝒙 𝟐 1 3 Make a table of values and graph the quadratic function y = x2. x y = x2 (x, y) 1 3 0 (0)2 = 0 (0, 0) 2 (2)2 = 1 (2, 1 ) 3 (3)2 = 3 (3, 3)

Ex 3 Comparing Widths of Parabolas Use the graphs below. Order the quadratic functions (x) = –x2, (x) = –3x2, and (x) = x2 from widest to narrowest graph. 1 2 (x) = –x2 (x) = –3x2 (x) = x2 1 2 Of the three graphs, (x) = x2 is the widest and (x) = –3x2 is the narrowest. So, the order from widest to narrowest is (x) = x2, (x) = –x2, (x) = –3x2. 1 2

Ex 4 Graphing 𝒚= 𝒂𝒙 𝟐 +𝒄 Graph the quadratic functions y = 3x2 and y = 3x2 – 2. Compare the graphs. x y = 3x2 y = 3x2 – 2 2 12 10 1 3 1 0 0 2 –1 3 1 The graph of y = 3x2 – 2 has the same shape as the graph of y = 3x2, but it is shifted down 2 units.

Make a table and graph the quadratic function 𝑓 𝑥 =−2 𝑥 2 Practice Make a table and graph the quadratic function 𝑓 𝑥 =−2 𝑥 2 Order the quadratic functions 𝑦= 𝑥 2 , 𝑦= 1 2 𝑥 2 , 𝑎𝑛𝑑 𝑦=−2 𝑥 2 in order widest to narrowest. 3. Graph 𝑦= 𝑥 2 and y= 𝑥 2 −4 x 𝑦=−2 𝑥 2 (x,y) -1 −2(−1) 2 =−2 (-1, -2) -2 (0) 2 =0 (0,0) 1 -2 (1) 2 =−2 (1, -2) Graph on the board 𝑦= 1 2 𝑥 2 , 𝑦= 𝑥 2 , 𝑦= −2𝑥 2 Graph on the board