Sketching Parabolas from Key Features

For y = x^2 - 4x + 3, does the parabola open upward or downward? What is its y-intercept?

It opens upward because a = 1 > 0. The y-intercept is (0, 3).

Find the vertex of y = x^2 - 4x + 3.

x = -(-4) / (2 * 1) = 2. y = (2)^2 - 4(2) + 3 = -1. The vertex is (2, -1).

Find the zeros of y = x^2 - 4x + 3 by factoring.

(x - 1)(x - 3) = 0, so the zeros are (1, 0) and (3, 0).

For y = -x^2 + 8x - 12, determine whether the vertex is a maximum or a minimum, and find its coordinates.

Because a = -1, it opens downward, so the vertex is a maximum. x = -8 / (2 * -1) = 4. y = -(4)^2 + 8(4) - 12 = 4. The vertex is (4, 4).

Find the axis intercepts for y = -x^2 + 8x - 12.

y-intercept is (0, -12). For x-intercepts: -x^2 + 8x - 12 = 0 simplifies to x^2 - 8x + 12 = 0, which factors to (x - 2)(x - 6) = 0. Intercepts are (2, 0) and (6, 0).

A company models profit with P = -2x^2 + 40x - 50, where x is items sold. How many items yield the maximum profit, and what is that profit?

Items needed: x = -40 / (2 * -2) = 10 items. Maximum profit: P = -2(10)^2 + 40(10) - 50 = 150 dollars.

Explain why the graph of y = x^2 + 4 has no x-intercepts, and identify its vertex.

Setting x^2 + 4 = 0 gives x^2 = -4, which has no real solutions (discriminant is 0^2 - 4(1)(4) = -16 < 0). The vertex is at (0, 4).