Desmos Problems

In Desmos, graph f(x) = −2x² + 12x + 5.

The parabola opens down because a = −2, so the vertex is a maximum.

The x-value of the vertex is

x = −b2a.

a = −2 and b = 12.

x = −122(−2).

x = −12−4.

x = 3.

f(3) = −2(3²) + 12(3) + 5.

f(3) = −18 + 36 + 5.

f(3) = 23.

Answer: 23

f(x) − 6 = x² + 5x.

Add 6 to both sides.

f(x) = x² + 5x + 6.

In Desmos, graph f(x). The parabola opens up, so the lowest point is a minimum.

The x-value of the vertex is

x = −b2a.

a = 1 and b = 5.

x = −52 · 1.

x = −52.

Desmos shows that same x-value as −2.5.

Answer: −52

Vertex form is g(x) = a(x − h)² + k.

The vertex is (h, k).

g(x) = 3(x − 2)² + 8.

h = 2 and k = 8.

In Desmos, the turning point of the graph is (2, 8).

Answer: (2, 8)

h(x) = (x + 1)(x − 7).

The x-intercepts are x = −1 and x = 7.

The axis of symmetry is halfway between the intercepts.

x = −1 + 72.

x = 62.

x = 3.

In Desmos, that is the vertical line through the vertex.

Answer: x = 3

f(x) = x² − 120x + 3618.

The y-intercept is 3618. The minimum is the y-value of the vertex.

In Desmos, graph f(x) and zoom until the lowest point is on the screen.

The x-value of the vertex is

x = −b2a.

a = 1 and b = −120.

x = −(−120)2 · 1.

x = 1202.

x = 60.

f(60) = 60² − 120(60) + 3618.

f(60) = 3600 − 7200 + 3618.

f(60) = 18.

Answer: 18

f(x) = 2x² − 16x.

Factor.

f(x) = 2x(x − 8).

The x-intercepts are the solutions of f(x) = 0.

2x = 0 or x − 8 = 0.

x = 0 or x = 8.

The intercepts are (0, 0) and (8, 0).

In Desmos, the graph crosses the x-axis at 0 and at 8.

Answer: 8