module 20.1 connecting intercepts and zeroes...module 20.1 connecting intercepts and zeroes how can...
TRANSCRIPT
Module 20.1
Connecting Intercepts And Zeroes
How can you use the graph of a quadratic functionto solve its related quadratic equation?
P. 937
As we said in Module 19.2 β Quadratic functions can take more than one form.
The first is called Vertex Form. Here it is: π π = π(π β π)π + πExample: π π = π(π β π)π + πWe learned how to graph a quadratic function in this form on page 908.
Now we focus on the second, called Standard Form.Here it is: π = πππ + ππ + πExample: π = πππ + ππ β π
How do we graph a quadratic function in this form?
One way is to create a table of x and y values, and then plot them.
π = πππ + ππ β π
How do you determine the axis of symmetry?
The axis of symmetry for a quadratic equation
in standard form is given by the equation π = βπ
ππ
So if we have the equation π = πππ + ππ β π
Then the axis of symmetry is βπ
ππ= β
π
π π= β
π
π= β1
Thatβs a vertical line with the equation π = βπ.
So we know the x-coordinate of the vertex ( β1),which is one half of the vertex.
How do you find the vertex?
Substitute the value of the axis of symmetry for π into the equation and solve for y.
π = πππ + ππ β π= π(βπ)π+π βπ β π= π π β π β π= π β π β π = βπ
So the vertex is at (β1, β7).
P. 938Just like there are quadratic functions, like π π = πππ + ππ + πThere are also quadratic equations, like πππ β π = βπ
How do you solve a quadratic equation?One way to do it is to factor it and find the βzeroesβ.Another way is to do it graphically.
Itβs a 5-Step process.
Step 1: Convert the equation into a βrelatedβ function by rewriting it so that it equals zero on one side.
πππ β π = βπ+ 3 + 3 Add 3 to both sides, so the right side will equal 0
πππ β π = π
Step 2: Replace the zero with a y.πππ β π = ππ² = πππ β π Re-order it
Step 3: Make a table of values for this βrelatedβ function.π² = πππ β π
Step 4: Plot the points and sketch the graph.
Step 5: The solution(s) of the equation are the x-intercepts, also known as the βzerosβ of thefunction. In this case theyβre π = π and π = βπ.
P. 938
A zero of a function is an x-value that makes the value of the function 0.
The zeros of a function are the x-intercepts of the graph of the function.
A quadratic function may have one, two, or no zeros.
P. 938
One Zero: π¦ = 2π₯2
When is 2π₯2 = 0 ?Only when π₯ = 0.
Two Zeros: π¦ = 2π₯2 β 2When is 2π₯2 β 2 = 0 ?When π₯ = β1 πππ π₯ = 1.
No Zeros: π¦ = 2π₯2 + 2When is 2π₯2 + 2 = 0 ?Never!
P. 941-942
You can solve this algebraically:Subtract 10 from both sides to get βππππ + ππ = π.Add ππππ to both sides, to get ππππ = ππ.Divide both sides by 16, so ππ = π. ππ.Take the square root of both sides to get π = Β±π. π.Since time canβt be negative, the answer has to be 1.5 seconds.
Or you can solve this graphically:
β16π‘2 + 36 = 0β16π‘2 + 36 = π¦
Create a table of x (or t) and y values, then graph those coordinates.
The y-axis represents the height, and the x-axis represents time.
When y=0, what is x (or t) ?