4.1 Graphical Solutions of Quadratic Equations
Chapter 4: Quadratic Equations:
4.1 Graphical Solutions of Quadratic Equations
Investigation:
Graph the following Quadratic Functions:
y = x2 - 4x - 5
y = x2 - 6x + 9
0 = x2 - 4x - 5
0 = x2 - 6x + 9
y = x2 + 1
0 = x2 + 1
Replacing zero for y changes each function into an equation.
How can you use the graphs above to solve each equation.
In general,
You can solve a quadratic equation of the form ax 2 + bx + c = 0 by graphing
the function f(x) = ax2 + bx + c .
The solutions to a quadratic equation are called the roots of the equation. You can
find the roots of the equation by determining the x-intercepts (or zeros) of the graph
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Ex. Solve the quadratic equation graphically.
-x2 + 5x - 4 = 0
Ex. Solve the quadratic equation graphically.
x2 - 8 = 0
Ex. Solve the quadratic equation graphically.
3x2 - x = -2
Note: You need to set all quadratic equations equal to zero before you solve. Why?
Use 3m2 - m = -2 to explain.
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Ex. Two consecutive numbers have a product of 6. Write a quadratic equation and solve graphically.
Note: Write each using algebra.
1) Two consecutive numbers
2) Two consecutive even numbers
3) Two consecutive odd numbers
Ex. The parabolic arc of a diver making a dive is represented by the formula h = -5.1t2 +10t + 5.
https://www.desmos.com/calculator
a) What does the constant of 5 represent?
b) What does
h = -5.1t2 +10t + 5 represent?
c) How would you find out when the diver hits the water?
d) When does the diver hit the water?
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d) When does the diver hit the water?
In summary, A quadratic equation has zero, one, or two real roots.
no real roots
two equal real roots
( one root)
two distinct real roots
The roots can be found by graphing the corresponding quadratic function and
finding the x-intercepts (zeros).
Assignment: Sec. 4.1 p215 #1, 2, 3ace, 4bc, 7, 8ab, 9, 11, 13*
For #3ce, graph without graphing tool. (ie. Use a table of values
or vertex form to graph)
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