Video: Identifying Graphs of Factorable Quadratic Equations

Which of the following graphs represents the equation 𝑦 = βˆ’2π‘₯Β² + 9π‘₯ βˆ’ 7? [A] Graph A [B] Graph B [C] Graph C [D] Graph D [E] Graph E

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Video Transcript

Which of the following graphs represents the equation 𝑦 equals negative two π‘₯ squared plus nine π‘₯ minus seven?

Our function 𝑓 of π‘₯ equals negative two π‘₯ squared plus nine π‘₯ minus seven is given in the general form π‘Žπ‘₯ squared 𝑏π‘₯ plus 𝑐. But before we do anything with the function, let’s consider each of these graphs. When we look at the graph of (A), it opens upward and has a vertex in the fourth quadrant. The graph of (B) opens downward and has a vertex in the second quadrant. (C) opens downward and has a vertex in the first quadrant. (D) opens downward and has a vertex in the first quadrant. And (E) opens upward and has a vertex in the fourth quadrant.

A good strategy to finding the graph would be to take our equation and find its vertex. When we have an equation in the general form, we find the vertex to be negative 𝑏 over two π‘Ž. That’s the π‘₯-coordinate. So we take negative nine over two times negative two, which is positive 2.25. This means the π‘₯-coordinate of our vertex will be located at 2.25. If we draw the line π‘₯ equals 2.25 on all five of our graphs, we see the in (A) that intersects the vertex and in (C) that intersects the vertex. On that basis, (B), (D), and (E) are not the graphs we’re looking for.

To find the 𝑦-coordinate of the vertex, we could plug in 2.25 into our equation. When we solve for that, we get 3.125. And the vertex for (C) is located at 3.125. We also notice that our π‘Ž-value is negative. And that means our graph must open downward. Option (A) opens upward and has a vertex in the wrong place. And so we can say that option (C) is the correct graph here.

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