DISCOVER: Vertex and x-Intercepts We know that the graph of...

DISCOVER: Vertex and x-Intercepts We know that the graph of the quadratic function $f(x)=(x-m)(x-n)$ is a parabola. Sketch a rough graph of what such a parabola would look like. What are the $x$-intercepts of the graph of $f ?$ Can you tell from your graph the $x$ -coordinate of the vertex in terms of $m$ and $n ?$ (Use the symmetry of the parabola.) Confirm your answer by expanding and using the formulas of this section.

Key Concepts

Vertex

The vertex of a parabola is its turning point, where it achieves its maximum or minimum value depending on the direction the parabola opens. The x-coordinate of the vertex is found by averaging the x-intercepts, resulting in (m + n) / 2, which directly reflects the symmetry of the graph.

Axis of Symmetry

The axis of symmetry is the vertical line that divides the parabola into two equal halves, reflecting its mirror-image property. In the case of a factored quadratic function with roots m and n, this axis has the equation x = (m + n) / 2, which is also the x-coordinate of the vertex.

Parabola

A parabola is the graph produced by a quadratic function. It is characterized by a symmetric, curved shape that opens upwards if the leading coefficient is positive and downward if it is negative. Its symmetry about a vertical line—termed the axis of symmetry—facilitates the determination of its key features.

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in forms such as f(x) = ax² + bx + c or in factored form. Its graph is a parabola, which makes the function useful for modeling various phenomena where a maximum or minimum value is significant.

x-intercepts

The x-intercepts are the points where the graph of the quadratic function crosses the x-axis. For a quadratic function given in factored form as f(x) = (x - m)(x - n), the x-intercepts are the values x = m and x = n, which represent the roots of the equation.

Transcript

00:01 So for this, i'm just going to go through some sort of relationships and aspects of what a parabola looks like in general.

00:10 So we have a general equation for a parabola here.

00:14 So we're going to sketch a rough graph of what this parabola would look like.

00:18 We already have our x intercepts.

00:22 So our two x intercepts, because if you imagine solving this equation here, you'll have x minus m equals zero, so x equals m, you know, x minus n equals zero.

00:35 So x equals n.

00:39 So our two x intercept here are going to be m and n.

00:46 And we can assume they are both positive for now, because we have a positive x squared as well.

00:50 So we can assume both of those are positive for now.

00:54 So n and m.

00:57 So those are going to be our x intercepts.

01:00 And now, we're going to try and work out our y intercept.

01:04 So if you imagine expanding this out, you'll get a y intercept of n times m because i'll expand this out.

01:12 So you'll get x squared minus mn, mx, sorry, minus n x, plus m times n.

01:26 So then if you imagine subbing in zero to try and get the y intercept, anything with an x and it's going to go to zero and you'll be left with m.

01:35 So the y intercept is actually going to be m and if we're going to try and draw this graph here you'll see that it will look something a little like that.

01:48 And now it's asking for us to try and estimate the value of our turning point.

01:53 So we have a minimum value here...

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