QUADRATIC FUNCTION

Intro… Functions with the form y=ax 2 +bx+c are called quadratic functions and their graphs have a parabolic shape When we solve ax 2 +bx+c=0 we look for values of x that are x-intercepts (because we have y=0) The x-intercepts are called the solutions or roots of a quadratic equation

Solving Quadratic Equations by Graphing Quadratic equation y=ax 2 +bx+c ax 2 is the quadratic term, bx is the linear term, and c is the constant term

A quadratic equation can have –two real solutions, –one real solution, –or no real solutions

Solving Quadratic Equations by Factoring Factor with the zero product property: if a*b=0 then either a=0 or b=0 or both are equal to 0 Factoring by guess and check is useful, but you may have to try several combinations before you find the correct one While doing word problems examine your solutions carefully to make sure it is a reasonable answer

The Quadratic Formula and the Discriminant The quadratic formula gives the solutions of ax 2 + bx + c = 0 when it is not easy to factor the quadratic or complete the square Quadratic formula: The b 2 – 4ac term is called the discriminant and it helps to determine how many and what kind of roots you see in the solution

Example Graph y= -x 2 - 2x + 8 and find its roots. Vertex: (-1, 9) Roots: (-4, 0) (2, 0) Viewing window: Xmin= -10 Xmax=10 Ymin= -10 Ymax= 10

POSSIBLE SHAPES

4 langkah menggambar kurva Step 1 Determine the basic shape. The graph has a U shape if a > 0, and an inverted U shape if a < 0. Step 2 Determine the y intercept. This is obtained by substituting x = 0 into the function, which gives y = c. Step 3 Determine the x intercepts (if any). These are obtained by solving the quadratic equation Step 4 Determine the vertex by finding the symmetry and substitute the value of the x symemtry

The axis of symmetry is a line that divides a parabola into two equal parts that would match exactly if folded over on each other The vertex is where the axis of symmetry meets the parabola The roots or zeros (or solutions) are found by solving the quadratic equation for y=0 or looking at the graph

example F(x) = -x x – 12 Gambar grafiknya: 4 langkah. 1. menentukan basic shape. Karena a < 0 maka INVERTED U SHAPE 2. intercept dg sumbu y (x = 0) maka y = -12. jadi grafik akan memotong y pada (0, -12) 3. selesaikan persamaan tsb / cari nilai x nya 4. cari sumbu tengahnya dan titik puncaknya

The axis of symmetry is a line that divides a parabola into two equal parts that would match exactly if folded over on each other The vertex is where the axis of symmetry meets the parabola The roots or zeros (or solutions) are found by solving the quadratic equation for y=0 or looking at the graph

Graph with definitions shown: Three outcomes for number of roots: One root:Two roots No roots:

Example -x 2 : quadratic term -2x: linear term 8: constant term Vertex: x=(-b/2a) x= -(-2/2(-1)) x= 2/(-2) x= -1 Solve for y: y= -x 2 -2x + 8 y= -(-1) 2 -(2)(-1) + 8 y= -(1) y= 9 Vertex is (-1, 9) For y= -x 2 -2x + 8 identify each term, graph the equation, find the vertex, and find the solutions of the equation.

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