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Mathematics

Essay,
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IGCSE may provide you with:

Vertex and another point

x-intercepts and a point

Vertex or intercepts with a = 1

Remember: y = a(x – h)² + k – this is known as the vertex form of a quadratic function.

a is the constant.

(h, k) is the coordinate of the vertex.

Vertex forms have their uses, as you should be aware of:

If a < 0, we have a function with a maximum.

If a > 0, we have a function with a minimum.

Important to know:

When the vertex form is y = a(x + h)² + k,

the coordinate of the vertex will be (-h, k) since the value of h must be negative for the original vertex form to change: (x – (-h)) + k = (x + h).

Likewise,

When the vertex form is y = a(x – h)² – k,

the coordinate of the vertex will be (h, -k) since the value of k must be negative for the original vertex form to change: a(x – h)² + (-k) = a(x – h)² – k.

GIVEN VERTEX AND ANOTHER POINT:

Find the equation of the quadratic equation given the vertex (2, -4) and the y-intercept of 8.

y = a(x – h)² + k

1st piece of information given: vertex (2, -4), therefore, h = 2 and k = -4.

Substitute,

y = a(x – 2)² – 4.

2nd piece of information given: y-intercept of 8, therefore, x = 0 and y = 8.

Substitute,

8 = a(0 – 2)² – 4

8 = a(-2)² – 4

8 = 4a – 4

8 + 4 = 4a – 4 + 4

4a = 12

a = 3

Therefore, the equation is y = 3(x – 2)² – 4 or 3x² – 12x + 8.

**GIVEN X-INTERCEPT AND A POINT:**

Find the quadratic equation with x-intercepts (-2, 0) and (2, 0) and passing through the point (1, 3).

y = ax² + bx + c

For this question, you have to find a, b, and c.

1st piece of information given: x = -2, y = 0, and x = 2, y = 0.

Substitute:

0 = 4a – 2b + c [1]

0 = 4a + 2b + c [2]

[1] – [2] = 0 = -4b

Therefore, b = 0.

2nd piece of information given: x = 1 and y = 3.

Substitute:

3 = a + b + c [Since b = 0]

c = 3 – a [3]

Substitute [3] into [1]:

0 = 4a + 3 – a

3a + 3 = 0

Therefore, a = -1.

Since a = -1 and b = 0,

We can substitute this information into [1] to find c:

0 = -4 + c

Hence, c = 4.

Consider y = ax² + bx + c,

y = -x² + 4.

**VERTEX OR INTERCEPTS WITH A = 1:**

Find the quadratic equation in the form y = a(x – b)² + c,

when the vertex is at (3, -1) and the value of a = 1.

Information given: a = 1, h = 3, and k = -1."

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