Lesson 10 – 1 Exploring Quadratic Graph
Standard form – y = ax2 + bx + c
Quadratic parent function (simplest quadratic function) – y = x2 or f (x) = x2
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 the same. The line that divides the parabola into two matching halves is called the axis of symmetry. The formula for the axis of symmetry is x = -b / 2a. Line of symmetry is the x-coordinate of the symmetry. The highest or lowest point of a parabola is its vertex. It is on the axis of symmetry.
y = ax2 + bx + c
If a is greater than zero; a > 0; the graph would turn out to be a regular U-shaped curve, and the vertex would be minimum.
If a is less than zero; a <> 0; the graph opens up. If a is less than zero; a < y =" –10x2" y =" –0.9x2" y =" 9x2" y =" x2" y =" –7x2," y =" –" y =" –" y =" –" y =" –" y =" –7x2" y =" –7x2," y =" –" y =" –" y =" –" y =" –7x2," y =" –" y =" –" y =" –" y =" –7x2">
Lesson 10 – 2 Quadratic Function
Role of “b” affects the position of the symmetry. It also moves the graph left or right. The equation of the axis of symmetry is x = -b / 2a. When graphing in standard form (learned in lesson 10 – 1) it gives you the middle value of the table y = ax2 + bx + c.
Graphing inequality is very similar to graphing linear inequality.
Inequality with signs greater than or less than you would graph the inequality with dotted lines; < , > dotted lines - - -.
Inequality with signs greater than or equal to or less than or equal to you would graph the inequality with solid lines; ≤ , ≥ solid line ───.
Shading inequality graphs. Inequality with signs greater than or greater than or equal to you would shade the upper part of the graph; >, ≥.
Inequality with signs less than or less than or equal to you would shade the lower part of the graph; < , ≤. Sample Questions: 1. Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = 2x2 – 13x + 18. x = ; vertex: ( , ) x = – ; vertex: (– , – ) x = – ; vertex: (– , 3) x = ; vertex: ( , – ) 2. Find the coordinates of the vertex of the graph of y = 4x2 + x + 3. ( , ) (–8, ) (– , ) (– , 3) 3. Match the graph with its function.
Standard form – y = ax2 + bx + c
Quadratic parent function (simplest quadratic function) – y = x2 or f (x) = x2
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 the same. The line that divides the parabola into two matching halves is called the axis of symmetry. The formula for the axis of symmetry is x = -b / 2a. Line of symmetry is the x-coordinate of the symmetry. The highest or lowest point of a parabola is its vertex. It is on the axis of symmetry.
y = ax2 + bx + c
If a is greater than zero; a > 0; the graph would turn out to be a regular U-shaped curve, and the vertex would be minimum.
If a is less than zero; a <> 0; the graph opens up. If a is less than zero; a < y =" –10x2" y =" –0.9x2" y =" 9x2" y =" x2" y =" –7x2," y =" –" y =" –" y =" –" y =" –" y =" –7x2" y =" –7x2," y =" –" y =" –" y =" –" y =" –7x2," y =" –" y =" –" y =" –" y =" –7x2">
Lesson 10 – 2 Quadratic Function
Role of “b” affects the position of the symmetry. It also moves the graph left or right. The equation of the axis of symmetry is x = -b / 2a. When graphing in standard form (learned in lesson 10 – 1) it gives you the middle value of the table y = ax2 + bx + c.
Graphing inequality is very similar to graphing linear inequality.
Inequality with signs greater than or less than you would graph the inequality with dotted lines; < , > dotted lines - - -.
Inequality with signs greater than or equal to or less than or equal to you would graph the inequality with solid lines; ≤ , ≥ solid line ───.
Shading inequality graphs. Inequality with signs greater than or greater than or equal to you would shade the upper part of the graph; >, ≥.
Inequality with signs less than or less than or equal to you would shade the lower part of the graph; < , ≤. Sample Questions: 1. Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = 2x2 – 13x + 18. x = ; vertex: ( , ) x = – ; vertex: (– , – ) x = – ; vertex: (– , 3) x = ; vertex: ( , – ) 2. Find the coordinates of the vertex of the graph of y = 4x2 + x + 3. ( , ) (–8, ) (– , ) (– , 3) 3. Match the graph with its function.
y = x2 + 5x
y = –x2 – 5
y = x2 + 5
y = –x2 – 5x
Lesson 10 – 3 Solving Quadratic Equations
Standard form of a quadratic equation; y = ax2 + bx + c; can have one, two or no-real number solution. The solution of a quadratic equation and the related x-intercept are often called roots of the equation.
When a quadratic equation doesn’t have a “b” value you use square roots to solve the equation. You solve the equation by finding the square roots of each side; y = ax2 , y = ax2 + c.
Sample Questions:
1. Solve a2 + 36 = 0 by finding square roots.
–8, 8
no solution
–12, 12
–6, 6
2. Find the side of a square with an area of 79 ft2. If necessary, round to the nearest tenth.
4.3 ft
8.9 ft
6,241 ft
39.5 ft
3. Solve x2 + 6 = 0.
–16, 16
–6, 6
–1, 1
no solution
Lesson 10 – 4 Factoring to Solve Quadratic Equation
Use factoring to solve a quadratic equation you will use one more step → tee off (use a T chart). You have to set the equation equal to zero, and make sure you set the standard form equation equal to zero.
Sample Question:
1. Use the Zero-Product Property to solve –2x(2x + 5) = 0.
2, –
0, –
2,
0, –
2. Solve 16x = x2 by factoring.
0, 16
–4, 4
0, 4
1, 16
3. Solve 15 = 8x2 – 14x.
– ,
– ,
–3,
–5,
y = –x2 – 5
y = x2 + 5
y = –x2 – 5x
Lesson 10 – 3 Solving Quadratic Equations
Standard form of a quadratic equation; y = ax2 + bx + c; can have one, two or no-real number solution. The solution of a quadratic equation and the related x-intercept are often called roots of the equation.
When a quadratic equation doesn’t have a “b” value you use square roots to solve the equation. You solve the equation by finding the square roots of each side; y = ax2 , y = ax2 + c.
Sample Questions:
1. Solve a2 + 36 = 0 by finding square roots.
–8, 8
no solution
–12, 12
–6, 6
2. Find the side of a square with an area of 79 ft2. If necessary, round to the nearest tenth.
4.3 ft
8.9 ft
6,241 ft
39.5 ft
3. Solve x2 + 6 = 0.
–16, 16
–6, 6
–1, 1
no solution
Lesson 10 – 4 Factoring to Solve Quadratic Equation
Use factoring to solve a quadratic equation you will use one more step → tee off (use a T chart). You have to set the equation equal to zero, and make sure you set the standard form equation equal to zero.
Sample Question:
1. Use the Zero-Product Property to solve –2x(2x + 5) = 0.
2, –
0, –
2,
0, –
2. Solve 16x = x2 by factoring.
0, 16
–4, 4
0, 4
1, 16
3. Solve 15 = 8x2 – 14x.
– ,
– ,
–3,
–5,
