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Which of the following graphs represents 𝑓 of 𝑥 equals 𝑥 squared plus one?
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Almost immediately, I notice that the leading coefficient here is positive.
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What do I mean by that?
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This 𝑥 squared is a positive number.
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You can imagine it as one times 𝑥 squared, so its coefficient is one, positive one.
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And why does that matter when we’re graphing quadratic equations?
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The leading coefficient with our 𝑥 squared will affect which direction our parabola opens.
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Positive values means our parabola opens upward; negative values means our parabola opens downward.
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Since our 𝑥 squared value is positive, our parabola will open upward.
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This means that both (b) and (d) are not possible options for graphs of this function.
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But now we need an additional point to help us figure out if (a) is our graph or (c) is our graph.
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To do that, I want to choose a point that we can see on our graph.
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Let’s plug in zero for 𝑥.
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If we plug in zero for 𝑥, we get zero 𝑥 squared plus one.
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This function at 𝑥 equals zero equals one.
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The point zero, one is on this parabola, so let’s graph it.
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There’s point zero, one on graph (a), and here’s point zero, one on graph (c).
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Zero, one is a point on both (a) and (c), so we’ll need to choose an additional value to check.
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Let’s choose another number.
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Let’s check the value of the function when 𝑥 equals one.
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That would be equal to one squared plus one.
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The value of this function when 𝑥 equals one is two.
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We found an additional point of one, two.
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Let’s graph that one.
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Here’s the point one, two on graph (a), and here’s the point one, two on graph (c).
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The point one, two doesn’t fall on the line in graph (c); we can eliminate it.
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Graph (a) represents the function 𝑓 of 𝑥 equals 𝑥 squared plus one.