Module 20.1

Connecting Intercepts And Zeroes

How can you use the graph of a quadratic functionto solve its related quadratic equation?

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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?

One way is to create a table of x and y values, and then plot them.

𝒚 = 𝟑𝒙𝟐 + 𝟔𝒙 − 𝟒

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. 937

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.

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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!

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P. 941

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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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