9.4 – Solving Quadratic Equations
BY GRAPHING!
Warm-Up
1. Solve for x in the equation
2. Graph
What is a Quadratic Equation?
A quadratic equation in standard form is written:
y = ax2 + bx + c, where a ≠ 0
Roots of a Quadratic Equation
• The roots of a quadratic equation are the solutions to:
0 = ax2 + bx + c
Quadratic equation in standard formwith y = 0
What kind of points on a graph have y-values of 0? Where do we find these points? What might we call them?
Roots of a Quadratic Equation
• Roots are represented graphically by the x-intercepts of the graph of a quadratic equation.
RootsRoots, Roots, Baby!
Connecting Solutions to Roots
Let’s look at the equation:
Solve for .
Connecting Solutions to Roots
Without doing any calculations solve , given the graph below of .
x = 2, -2
Quick Practice!
Put the following equation into standard form:
Quick Practice!
Below is the graph of , determine the roots and check them algebraically.
Quick Practice!
Put the following equation into standard form:
Quick Practice!
Below is the graph of , determine the roots and check them algebraically.
Quick Practice!
Put the following equation into standard form:
Quick Practice!
Below is the graph of , determine the roots and check them algebraically.
So how does this help me?
Using graphing to solve:
1. Rewrite the equation into standard form.2. Change the 0 into a y and graph the
equation: .3. Identify the roots of the equation, which are
your solutions to the original equation.4. Check to see if these work algebraically!
I GET IT NOW!!!
Getting it Done by Hand
Solve the following equation by graphing (you may not use any graphing technology):
𝑥2−𝑥=2
x -1 0 1 2
y
Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=
button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),
setting left and right boundaries and making a guess. 5. Check answers!
Graph: 4x2 = 16
Using a Calculator
Using a Calculator
Graph: x2 - 4x = 5
Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=
button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),
setting left and right boundaries and making a guess. 5. Check answers!
Using a Calculator
Graph: x2 = -x + 6
Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=
button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),
setting left and right boundaries and making a guess. 5. Check answers!
Using a Calculator
Graph: Make one up!
Steps to graph a quadratic equation:1. Put equation into standard form.2. Replace the 0 with y.3. Graph the function on your calculator using the Y=
button.4. Find the zeros using the “CALC” menu ( 2nd TRACE ),
setting left and right boundaries and making a guess. 5. Check answers!
A baseball is thrown at 100 mph ft/sec from left field toward home plate. The models below give paths of the ball for two initial angles, with height of y and horizontal distance x (both measure in feet)
o 2
o 2
15 : y .00171x .268x 6
25 : y .00195x .466x 6
If home plate is 236 feet away, which angle(s) have the ball hitting the ground before reaching the plate?
Homework
Complete worksheet by tomorrow!
Quiz 9.4-9.6 on Monday 5/6!