Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of

Write the quadratic function in standard form and sketch its...

Key Concepts

X-intercepts

The x-intercepts are the points where the quadratic function crosses the x-axis, found by setting f(x) = 0 and solving for x. They indicate the values of x for which the quadratic equals zero, and the number of x-intercepts depends on the discriminant of the quadratic equation.

Vertex

The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value. In the vertex form f(x) = a(x - h)^2 + k, the vertex is located at (h, k). It represents the turning point of the parabola.

Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex of the parabola, dividing it into two congruent halves. In the vertex form, its equation is x = h, where h is the x-coordinate of the vertex.

Standard Form

The standard form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easier to identify important features of the graph, such as the vertex and the axis of symmetry.

Quadratic Function

A quadratic function is a second-degree polynomial, typically represented by f(x) = ax^2 + bx + c, where a, b, and c are constants and a ? 0. Its graph is a parabola which opens upward if a is positive and downward if a is negative.

Completing the Square

Completing the square is a method used to convert a quadratic function from its general form to its vertex form. This technique helps in rewriting the expression as a perfect square trinomial plus or minus a constant, thereby revealing the vertex of the parabola.

Transcript

00:01 Okay, so here we got a function f of x equals x squared plus 3x plus 1 fourth, and we're trying to get it into standard form, which resembles m of x above it.

00:10 So i'm going to take a look at these terms here, x squared plus 3x, and try to pick out a perfect square that is going to match those two terms when it's expanded out.

00:23 And so for me, that is x plus three halves squared.

00:31 Now, to see why, we can foil this out.

00:34 We have x squared plus two times three over two, x, plus three halves squared, which is nine -fourths.

00:50 So these two's canceled, and we're left with x squared plus three -x.

00:53 Now what we want is to have 9 force over here.

00:56 So what we're going to have to do to this equation above is add 8 fourths, which is 2.

01:04 But so let's get rid of this over here.

01:10 And so add 8 fourths to both sides.

01:12 What we have is f of x plus 8 over 4 is equal to x plus 3 quantity squared plus plus 0 now.

01:26 But until we move this plus a force cover, so i'm just going to subtract that from both sides and there we have our function in standard form now the good thing about standard form is that it's easy to get our vertex as hk so in this case it's negative three halves negative eight fourths be careful with the signs because remember in the standard form it's x minus h, but here we have plus but the h is actually negative and and so that gives us our axis of symmetry at x equals negative three halves.

02:01 And i believe all we do now is the intercepts.

02:04 So intercepts are the points where the value of the function f of x equals zero because it lands on the x -axis here.

02:11 So i'm going to take that to be zero and just solve what we have here.

02:20 I'm going to move this eight -fourths over.

02:23 X plus three halves, square, be squared.

02:27 And now i'm going to unsquare this entire thing...

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