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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 โ Quadratic functions can take more than one form.
The first is called Vertex Form. Here it is: ๐ ๐ = ๐(๐ โ ๐)๐ + ๐Example: ๐ ๐ = ๐(๐ โ ๐)๐ + ๐We 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: ๐ = ๐๐๐ + ๐๐ + ๐Example: ๐ = ๐๐๐ + ๐๐ โ ๐
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.
๐ = ๐๐๐ + ๐๐ โ ๐
How do you determine the axis of symmetry?
The axis of symmetry for a quadratic equation
in standard form is given by the equation ๐ = โ๐
๐๐
So if we have the equation ๐ = ๐๐๐ + ๐๐ โ ๐
Then the axis of symmetry is โ๐
๐๐= โ
๐
๐ ๐= โ
๐
๐= โ1
Thatโs a vertical line with the equation ๐ = โ๐.
So we know the x-coordinate of the vertex ( โ1),which is one half of the vertex.
How do you find the vertex?
Substitute the value of the axis of symmetry for ๐ into the equation and solve for y.
๐ = ๐๐๐ + ๐๐ โ ๐= ๐(โ๐)๐+๐ โ๐ โ ๐= ๐ ๐ โ ๐ โ ๐= ๐ โ ๐ โ ๐ = โ๐
So the vertex is at (โ1, โ7).
P. 938Just like there are quadratic functions, like ๐ ๐ = ๐๐๐ + ๐๐ + ๐There are also quadratic equations, like ๐๐๐ โ ๐ = โ๐
How do you solve a quadratic equation?One way to do it is to factor it and find the โzeroesโ.Another way is to do it graphically.
Itโs a 5-Step process.
Step 1: Convert the equation into a โrelatedโ function by rewriting it so that it equals zero on one side.
๐๐๐ โ ๐ = โ๐+ 3 + 3 Add 3 to both sides, so the right side will equal 0
๐๐๐ โ ๐ = ๐
Step 2: Replace the zero with a y.๐๐๐ โ ๐ = ๐๐ฒ = ๐๐๐ โ ๐ Re-order it
Step 3: Make a table of values for this โrelatedโ function.๐ฒ = ๐๐๐ โ ๐
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 โzerosโ of thefunction. In this case theyโre ๐ = ๐ and ๐ = โ๐.
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: ๐ฆ = 2๐ฅ2
When is 2๐ฅ2 = 0 ?Only when ๐ฅ = 0.
Two Zeros: ๐ฆ = 2๐ฅ2 โ 2When is 2๐ฅ2 โ 2 = 0 ?When ๐ฅ = โ1 ๐๐๐ ๐ฅ = 1.
No Zeros: ๐ฆ = 2๐ฅ2 + 2When is 2๐ฅ2 + 2 = 0 ?Never!
P. 941-942
You can solve this algebraically:Subtract 10 from both sides to get โ๐๐๐๐ + ๐๐ = ๐.Add ๐๐๐๐ to both sides, to get ๐๐๐๐ = ๐๐.Divide both sides by 16, so ๐๐ = ๐. ๐๐.Take the square root of both sides to get ๐ = ยฑ๐. ๐.Since time canโt be negative, the answer has to be 1.5 seconds.
Or you can solve this graphically:
โ16๐ก2 + 36 = 0โ16๐ก2 + 36 = ๐ฆ
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) ?