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Let's work through some final problems.

Click on each tab to practice graphing the quadratic equations and finding the quadratic equations from the graphs.

Problem 1

Problem 2

Problem 3

Problem 4


Graph the equation \(\small\mathsf{ f(x) = \frac{1}{2}(x-1)^{2} - 5 }\).

The vertex is at (h, k) = (1, -5).

The parabola opens up and is wider than the parent function, \(\small\mathsf{ a = \frac{1}{2} < 1 }\).

It goes through the points (-1, -3) and (3, -3).

graph 1


Graph the equation \(\small\mathsf{ f(x) = -2(x+3)^{2} + 5 }\).

The vertex is at (h, k) = (-3, 5).

The parabola opens down and is narrower than the parent function, \(\small\mathsf{ |a| = 2 > 1 }\).

It goes through the points (-4, 3) and (-2, 3).

graph 2


Find the equation of the graph.

graph 3

The vertex is at (h, k) = (2, 1).

The parabola opens up, so a > 0.

It goes through the point (1, 2).

\(\small\mathsf{ f(x) = a(x-2)^{2} + 1 }\)
\(\small\mathsf{ f(1) = a(1-2)^{2} + 1 }\)
\(\small\mathsf{ 2 = a(-1)^{2} + 1 }\)
\(\small\mathsf{ 2 = 1a + 1 }\)
\(\small\mathsf{ a = 1 }\)
\(\small\mathsf{ f(x) = 1(x-2)^{2} + 1 }\)

Find the equation of the graph.

graph 4

The vertex is at (h, k) = (-3, 6).

The parabola opens down, so a < 0.

It goes through the point (-4, 4).

\(\small\mathsf{ f(x) = a(x+3)^{2} + 6 }\)
\(\small\mathsf{ f(-4) = a(-4+3)^{2} + 6 }\)
\(\small\mathsf{ 4 = a(-1)^{2} + 6 }\)
\(\small\mathsf{ 4 = 1a + 6 }\)
\(\small\mathsf{ a = -2 }\)
\(\small\mathsf{ f(x) = -2(x+3)^{2} + 6 }\)