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 graph of the function crosses the x-axis, which occur when f(x) = 0. These intercepts can be found by solving the quadratic equation ax² + bx + c = 0, often using the quadratic formula, and they indicate where the function evaluates to zero.

Quadratic Functions

Quadratic functions are polynomial functions of degree two that can be expressed in the form ax² + bx + c, where a, b, and c are constants and a is non-zero. They generate parabolic graphs that either open upward or downward, depending on the sign of the coefficient a.

Standard Form

The standard form of a quadratic function is written as f(x) = ax² + bx + c. This form is particularly useful for identifying the y-intercept directly and serves as a starting point for converting the function into vertex form through the process of completing the square.

Vertex

The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. It can be found using the formula (-b/(2a), f(-b/(2a))) when the function is in standard form, and it serves as a critical point in graphing the parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, splitting it into two mirror-image halves. Its equation is given by x = -b/(2a), which helps in analyzing the parabola’s symmetrical properties.

Transcript

00:01 Okay, this question asks us to take a function f of x and get it into standard form, much like m of x above it.

00:08 So the first thing i notice is that there's a two in front of this equation, and in front of the x, the x square term in this equation.

00:15 So i'm going to factor that out, and that leaves us with two times the quantity of x squared minus a half x plus a half.

00:22 Now, if you notice that when we distribute this two, those fractions are going to go away.

00:27 We'll be left with exactly what we had.

00:28 Now, these first two terms are the same as the first two terms of x minus the fourth quantity squared, right? but the constant term at the end of this one, you expanded out, will be plus 116th.

00:45 But here we have a half, right? and so what i'm going to do is i'm going to subtract 7 eighths from both sides.

00:56 Now, 7 .a's doesn't exactly seem like what would get us to being, you know, this 116th.

01:05 But we have to remember that we have this 2 being distributed there.

01:10 And so everything works out that way.

01:12 And so we've got f of x minus 7 over 8 is equal to 2 times x minus 1 4th quantity squared.

01:22 And we just have to move that 7 eighths back over.

01:26 We've got two times of x minus.

01:34 Just to confirm that that 7 over 8 was correct, we can go ahead and distribute this out, and i'll leave the 2 out, actually.

01:44 And we'll have x squared minus 1⁄2x, and then we'll have that plus 1 over 16 that we talked about, and 7 .8s, which we can rewrite as 14 over 16.

01:57 Now, recall this 2 distributes there, so we have two 16ths there, so 216 plus the 1416s is the one that we had before.

02:10 So this is a standard form right here, which makes it easy to find a vertex as hk, so that will be 1 4th, 7, 8s, and our axis of symmetry is then 1x equal 1 4th...

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