Math

Question Determine the direction of opening for the quadratic equation y=(x3)(x2)y=(x-3)(x-2).

Studdy Solution

STEP 1

Assumptions
1. The given equation is in the form y=(xa)(xb)y = (x - a)(x - b), which is a quadratic equation.
2. The standard form of a quadratic equation is y=ax2+bx+cy = ax^2 + bx + c.
3. The direction of opening of a parabola (graph of a quadratic equation) is determined by the coefficient of the x2x^2 term.
4. If the coefficient of x2x^2 is positive, the parabola opens upwards.
5. If the coefficient of x2x^2 is negative, the parabola opens downwards.

STEP 2

We need to expand the given quadratic equation to identify the coefficient of the x2x^2 term.
y=(x3)(x2)y = (x - 3)(x - 2)

STEP 3

Use the distributive property (FOIL method) to expand the equation.
y=x22x3x+6y = x^2 - 2x - 3x + 6

STEP 4

Combine like terms to simplify the equation.
y=x25x+6y = x^2 - 5x + 6

STEP 5

Now that we have the quadratic equation in standard form, we can identify the coefficient of the x2x^2 term.
a=1a = 1

STEP 6

Since the coefficient of the x2x^2 term (aa) is positive, we can determine the direction of the opening of the parabola.
a>0The parabola opens upwards.a > 0 \Rightarrow \text{The parabola opens upwards.}
The direction of opening of the parabola represented by the equation y=(x3)(x2)y=(x-3)(x-2) is Up.

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