10. Factoring is an algebraic process, which is an essential one in the field of mathematics. Consider the following equation:
𝑦 = 𝑥2 + 2𝑥 − 8
This equation in factored form can be written as:
𝑦 = (𝑥 − 2)(𝑥 + 4)
Explain using words and algebra, how the zeros, the axis of symmetry, and the
vertex can be obtained.
The zeros of an equation are the values of x that make y equal to zero. In other words, they are the values of x where the graph of the equation intersects the x-axis. To find the zeros of the equation 𝑦 = (𝑥 − 2)(𝑥 + 4), we can set y equal to zero and solve for x:
0 = (𝑥 − 2)(𝑥 + 4)
This equation is true if and only if either 𝑥 − 2 = 0 or 𝑥 + 4 = 0. Solving each equation gives us:
𝑥 − 2 = 0 → 𝑥 = 2
𝑥 + 4 = 0 → 𝑥 = −4
Therefore, the zeros of the equation are x = 2 and x = −4.
The axis of symmetry of the graph is a vertical line that divides the graph into two symmetric halves. For a quadratic equation in standard form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, the axis of symmetry is the line given by x = −𝑏/2𝑎. In the case of 𝑦 = (𝑥 − 2)(𝑥 + 4), we can rewrite it in standard form:
𝑦 = 𝑥2 + 2𝑥 − 8
From here, we can see that 𝑎 = 1, 𝑏 = 2, and 𝑐 = −8. Therefore, the axis of symmetry is:
x = −𝑏/2𝑎 = −2/2(1) = −1
The vertex of the graph is the point where the curve reaches its highest or lowest point. For a quadratic equation in standard form, the vertex is given by the point (−𝑏/2𝑎, 𝑐−𝑏2/4𝑎). In the case of 𝑦 = (𝑥 − 2)(𝑥 + 4), we can rewrite it in standard form:
𝑦 = 𝑥2 + 2𝑥 − 8
From here, we can see that 𝑎 = 1, 𝑏 = 2, and 𝑐 = −8. Therefore, the vertex is:
( −𝑏/2𝑎, 𝑐−𝑏2/4𝑎 ) = ( −1, −9 )
Therefore, the zeros of the equation are x = 2 and x = −4, the axis of symmetry is x = −1, and the vertex is (−1, −9).
To find the zeros, axis of symmetry, and vertex of a quadratic equation in factored form, we can use the following steps:
1. Zeros: The zeros of a quadratic equation are the values of x for which y becomes zero. In factored form, the equation (y = (x - a)(x - b)) has zeros at x = a and x = b. In this case, the factored form is y = (x - 2)(x + 4), so the zeros are x = 2 and x = -4.
2. Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola into two equal halves. For a quadratic equation in factored form, the axis of symmetry is given by the average of the zeros. In this case, the average of the zeros is (2 + (-4))/2 = -1, so the equation has an axis of symmetry at x = -1.
3. Vertex: The vertex is the point on the parabola where it reaches its maximum or minimum value. For a quadratic equation in factored form, the vertex can be found by calculating the x-coordinate of the vertex using the axis of symmetry formula (x = -b/2a), and then substituting that value back into the equation to find the corresponding y-coordinate. In this case, the equation is y = (x - 2)(x + 4), so a = 1, b = 2, and c = -8. Substituting these values into the axis of symmetry formula, we get x = -2/2(1) = -1. Then, substituting x = -1 into the equation, we get y = (-1 - 2)(-1 + 4) = (-3)(3) = -9. Therefore, the vertex is (-1, -9).
In summary, for the given equation y = x^2 + 2x - 8 in factored form (y = (x - 2)(x + 4)), the zeros are x = 2 and x = -4, the axis of symmetry is x = -1, and the vertex is (-1, -9).