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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 – Quadratic functions can take more than one form.
The first is called Vertex Form. Here it is: 𝒈 𝒙 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌Example: 𝒈 𝒙 = 𝟑(𝒙 − 𝟐)𝟐 + 𝟒We 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: 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄Example: 𝒚 = 𝟑𝒙𝟐 + 𝟔𝒙 − 𝟒
How do we graph a quadratic function in this form?
How do you determine the axis of symmetry?
The axis of symmetry for a quadratic equation
in standard form is given by the equation 𝒙 = −𝒃
𝟐𝒂
So if we have the equation 𝒚 = 𝟑𝒙𝟐 + 𝟔𝒙 − 𝟒
Then the axis of symmetry is −𝒃
𝟐𝒂= −
𝟔
𝟐 𝟑= −
𝟔
𝟔= –1
That’s a vertical line with the equation 𝒙 = −𝟏.
So we know the x-coordinate of the vertex ( –1),which is one half of the vertex.
How do you find the vertex?
Substitute the value of the axis of symmetry for 𝒙 into the equation and solve for y.
𝒚 = 𝟑𝒙𝟐 + 𝟔𝒙 − 𝟒= 𝟑(−𝟏)𝟐+𝟔 −𝟏 − 𝟒= 𝟑 𝟏 − 𝟔 − 𝟒= 𝟑 − 𝟔 − 𝟒 = −𝟕
So the vertex is at (–1, –7).
P. 938Just like there are quadratic functions, like 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄There are also quadratic equations, like 𝟐𝒙𝟐 − 𝟓 = −𝟑
How do you solve a quadratic equation?One way to do it is to factor it and find the “zeroes”.Another way is to do it graphically.
It’s a 5-Step process.
Step 1: Convert the equation into a “related” function by rewriting it so that it equals zero on one side.
𝟐𝒙𝟐 − 𝟓 = −𝟑+ 3 + 3 Add 3 to both sides, so the right side will equal 0
𝟐𝒙𝟐 − 𝟐 = 𝟎
Step 2: Replace the zero with a y.𝟐𝒙𝟐 − 𝟐 = 𝒚𝐲 = 𝟐𝒙𝟐 − 𝟐 Re-order it
Step 3: Make a table of values for this “related” function.𝐲 = 𝟐𝒙𝟐 − 𝟐
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 “zeros” of thefunction. In this case they’re 𝒙 = 𝟏 and 𝒙 = −𝟏.
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: 𝑦 = 2𝑥2
When is 2𝑥2 = 0 ?Only when 𝑥 = 0.
Two Zeros: 𝑦 = 2𝑥2 − 2When is 2𝑥2 − 2 = 0 ?When 𝑥 = −1 𝑎𝑛𝑑 𝑥 = 1.
No Zeros: 𝑦 = 2𝑥2 + 2When is 2𝑥2 + 2 = 0 ?Never!
P. 941-942
You can solve this algebraically:Subtract 10 from both sides to get −𝟏𝟔𝒕𝟐 + 𝟑𝟔 = 𝟎.Add 𝟏𝟔𝒕𝟐 to both sides, to get 𝟏𝟔𝒕𝟐 = 𝟑𝟔.Divide both sides by 16, so 𝒕𝟐 = 𝟐. 𝟐𝟓.Take the square root of both sides to get 𝒕 = ±𝟏. 𝟓.Since time can’t be negative, the answer has to be 1.5 seconds.
Or you can solve this graphically:
−16𝑡2 + 36 = 0−16𝑡2 + 36 = 𝑦
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) ?
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