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If the point nine, zero is the vertex of the graph of the quadratic function π, what is the solution set of the equation π of π₯ is equal to zero?
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In this question, weβre given some information about a quadratic function π.
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Weβre told that the point nine, zero is the vertex of the graph of this function.
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We need to use this information to determine the solution set of the equation π of π₯ is equal to zero.
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And to do this, letβs start by recalling what we mean by the vertex of a quadratic function π.
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We can start by recalling the graph of any quadratic function will have a parabolic shape, and there are two possible orientations for this parabolic arc to take.
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First, when the leading coefficient is negative, we know the parabolic arc will open downwards.
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Second, if the leading coefficient is positive, then we know the parabolic arc will open upwards.
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So, these are the two possible general shapes that the graph of the function π can take.
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And thereβs something interesting we can note about both of these shapes.
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Thereβs a single turning point; we call this the vertex of the quadratic.
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And thereβs a few interesting properties worth noting about the vertex of a parabola.
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First, the line of symmetry of this parabola passes through the vertex.
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Second, if the quadratic opens downwards, then the vertex occurs at the maximum output of the function.
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And similarly, if the quadratic opens upwards, then the vertex occurs at the minimum output of the function.
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And we can use this to determine information about the graph of the function π.
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However, we also need to link this to the solution set of the equation π of π₯ is equal to zero.
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And we can do this by noting if π₯ is a solution to this equation, so π evaluated at π₯ is equal to zero, the output of the function is zero.
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So, the π¦-coordinate of this point on the curve will be zero.
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It will be an π₯-intercept of the graph.
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Therefore, we can answer this question by determining the possible π₯-intercepts of the graph of our function π of π₯ by knowing its vertex is the point nine, zero.
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So, letβs now sketch some possible graphs of the function π¦ is equal to π of π₯.
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Weβll start by marking the vertex on our graph; itβs the point nine, zero.
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One possible graph of the function π¦ is equal to π of π₯ might look like the following; itβs an upward-opening parabola with vertex at the point nine, zero.
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However, we do need to know there are infinitely many quadratics with vertex at the point nine, zero.
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For example, the graph of the function could also look like the following.
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It could be a wider parabola; it could also be a narrower parabola.
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Similarly, since weβre given no information about the function π of π₯ other than the coordinates of its vertex, the parabola could also open downwards.
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However, if we consider the π₯-intercepts of all of these possible graphs, we can notice something interesting.
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All of these graphs only have a single π₯-intercept, the vertex of the parabola.
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And this fact ties in with the properties of parabolas we discussed previously.
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In the parabola which opens downwards, the vertex occurs at the maximum output of the function, and in a parabola which opens upwards, the vertex occurs at the minimum output of the function.
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So, in both cases, we only get a single π₯-intercept to the point nine, zero.
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Therefore, thereβs only one solution to this equation; our value of π₯ needs to be equal to nine.
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And to write this as a solution set, this is the set of all possible solutions to the equation, and this gives us the set whose only element is nine.
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And itβs worth noting this property holds in general for any parabola.
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If the quadratic has a vertex which lies on the π₯-axis, then this is the only solution to the equation π of π₯ is equal to zero.
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It will have a single π₯-intercept.
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Therefore, we were able to show if nine, zero is the vertex of the graph of a quadratic function π, then the solution set of the equation π of π₯ is equal to zero is the set whose only element is nine.