Question Video: Determining a Quadratic Equation in Factored Form from its Graph | Nagwa Question Video: Determining a Quadratic Equation in Factored Form from its Graph | Nagwa

# Question Video: Determining a Quadratic Equation in Factored Form from its Graph Mathematics • Second Year of Secondary School

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### Video Transcript

To do this, letβs consider some features of the graph of quadratic functions like shape, π¦-intercept, roots, and vertex. The shape of this parabola opens upward. And that tells us that our π-value will be positive, greater than zero. We have a π¦-intercept located at the point zero, zero. To find the roots, we look for the place where the graph crosses the π₯-axis. Here we have a root at zero, zero and at five, zero. The vertex of this graph has been labeled. Itβs a minimum coordinate located at two and a half, negative 6.25.

What we wanna do now is think a little bit more about the roots. The roots are sometimes called solutions. Theyβre the place where our function equals zero. We have solutions when π₯ equals zero and when π₯ equals five. We can take these solutions and turn them into factors. Because π₯ is the same thing as π₯ minus zero, π₯ equals zero is a factor. And for factor two, we would want to say π₯ minus five equals zero. Here are two factors. However, we still donβt know what our π-value is because there are many equations that have the factors π₯ and π₯ minus five. And that means we should have π of π₯ equals π times π₯ times π₯ minus five.

To solve for π, we can plug in coordinates of points we know fall on this graph. We know that when π₯ equals 2.5, π of π₯ equals negative 6.25. Two and a half minus five is negative two and a half. Two and a half times negative two and a half equals negative 6.25. And if we divide both sides of the equation by negative 6.25, we see that π equals one. And if π equals one, then our factored form is π of π₯ equals π₯ times π₯ minus five. We always want to check for this π-value because as Iβve just drawn on top of this graph β hereβs an equation that has the same factors. However, in this case, the π would have to be less than zero because the graph opens downward, which means itβs not enough just to know the factors. You need to also check for the π-value.

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