Mastering Quadratic Functions: Formulas, Solutions, and Graphs

1. QUADRATIC FUNCTION

2. Intro… • Functions with the form y=ax2+bx+c are called quadratic functions and their graphs have a parabolic shape • When we solve ax2+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

3. Solving Quadratic Equations by Graphing • Quadratic equation y=ax2+bx+c • ax2 is the quadratic term, bx is the linear term, and c is the constant term

4. A quadratic equation can have • two real solutions, • one real solution, • or no real solutions

5. 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

6. The Quadratic Formula and the Discriminant • The quadratic formula gives the solutions of ax2 + bx + c = 0 when it is not easy to factor the quadratic or complete the square • Quadratic formula: • The b2 – 4ac term is called the discriminant and it helps to determine how many and what kind of roots you see in the solution

7. Example Graph y= -x2 - 2x + 8 and find its roots. Vertex: (-1, 9) Roots: (-4, 0) (2, 0) Viewing window: Xmin= -10 Xmax=10 Ymin= -10 Ymax= 10

8. POSSIBLE SHAPES

9. 4 langkahmenggambarkurva • 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

10. 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

11. example • F(x) = -x2 + 8 x – 12 • Gambargrafiknya: 4 langkah. • 1. menentukan basic shape. Karena a < 0 maka INVERTED U SHAPE • 2. intercept dg sumbu y (x = 0) maka y = -12. jadigrafikakanmemotong y pada (0, -12) • 3. selesaikanpersamaantsb / carinilai x nya • 4. carisumbutengahnyadantitikpuncaknya

13. 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

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

15. Example -x2: quadratic term -2x: linear term 8: constant term Vertex: x=(-b/2a) x= -(-2/2(-1)) x= 2/(-2) x= -1 For y= -x2 -2x + 8 identify each term, graph the equation, find the vertex, and find the solutions of the equation. Solve for y: y= -x2 -2x + 8 y= -(-1)2 -(2)(-1) + 8 y= -(1) + 2 + 8 y= 9 Vertex is (-1, 9)

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