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Dmitrij [34]
3 years ago
11

Find area of the isosceles triangle formed by the vertex and the x-intercepts of parabola y=x2+4x−12.

Mathematics
1 answer:
xz_007 [3.2K]3 years ago
6 0

The area of the isosceles triangle is 64 sq units.

<u>Solution:</u>

Part 1: x-intercepts

The x-intercepts occur at the points on the function where y=0

So, we need to solve

x^2-4x-12=0

The left side factors fairly easily into:

(x-6)(x+2)=0

So solution occur when

x-6=0\rightarrow x=6

and

x+2=0\rightarrow x=(-2)

So the x-intercepts are at (0,6) and (0,−2)

Part 2: vertex of the parabola

The vertex of a simple quadratic parabola occurs when the derivative of the quadratic is equal to 0.

The derivative of the given quadratic is

\frac{dy}{dx}=2x-4

By observation, this is equal to 0 when x=2

When x=2 the original equation becomes

y=(2)^2-4(2)-12

y=-16

Therefore the vertex of this parabola is at (2,−16)

The endpoints of the base of the isosceles triangle are (-6, 0) and (2, 0)

\Rightarrow so its base is 8

The height of the triangle reaches from the midpoint of the base (-2, 0) and the vertex (2, -16)

\Rightarrow so its height is 16

The area is  \frac{1}{2}\times \text { base }\times \text { height }=\frac{1}{2}\times8\times16=64 \text{ sq units }

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The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the distance of x from zero. This is why absolute value is never negative; absolute value only asks "how far?", not "in which direction?" This means not only that | 3 | = 3, because 3 is three units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero. You can see this on the following number line:

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Side note: When typing math as text, such as in an e-mail, the "pipe" character is usually used to indicate absolute values. The "pipe" is probably a shift-key somewhere north of the "Enter" key on your keyboard. While the "pipe" denoted on the physical keyboard key may look like a "broken" line, the typed character should display on your screen as a solid vertical bar. If you cannot locate a "pipe" character, you can use the "abs()" notation instead, so that "the absolute value of negative 3" would be typed as "abs(–3)".

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Simplify | 5 – 2 |.

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Simplify | 2 – 5 |.

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Simplify | 0(–4) |.

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In the next three examples, pay particular attention to the difference that the location of the square makes, with respect to the "minus" signs.

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