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In Exercises 17-34, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x-intercept(s).

$ h(x) = 4x^2 - 4x + 21 $

Vertex: $\left(\frac{1}{2}, 20\right)$Axis of symmetry: $x=\frac{1}{2}$$x$ -intercepts: None

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

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Harvey Mudd College

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Let's go ahead and write this in the vortex form. So here, let me start off by pulling out of four from the first two terms. So here we see the number in front of exes minus one. Divide that by two and then swear it. Good, but notice we didn't really add one over four because of this four hour here. So we added four times that. So we'll make up for it by subtracted. All right, now complete the square and we can see that r ver ticks is the point one half comma twenty? Yeah. And we think that letter just plug in a few more points here if you try half of one So that will be four times half square plus twenty, which is twenty one Similarly f of zero is also twenty one. Now let's try f i'll say one and a half What? Similarly, if you do f of negative a half So if we plug in negative a half will also get twenty four and we can keep plotting points and so on he triumph of two. So that will give you the force cancel so that I'll give you twenty nine And if you plug in negative one, that also could be twenty nine. And I always kind of have an idea of what our raft looks like. Okay, so that completes the sketch identifying the vertex Well, we already know what that is from the first tax form access of symmetry. This is always given by setting X equal to H. So in our case, the beach was one half. So this is the equation for the access of symmetry. And it's this blue line here vertical line passing through one half on the X axis and eccentricities. Well, we can see that there are no ex intercepts. You could see that graphically. But you could also see an algebraic Lee by setting this equal to zero and then solving for X. And you can see that there's no solution here because you can't take a square of a negative number. In other words, the right hand side is negative. But the left hand side is bigger than our equal to zero because it's a square. And this is why there are no ex intercepts

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