E ( C The axis of symmetry is a vertical line drawn through the vertex. a , which means the nth iteration of The vertex also has [latex]x[/latex] coordinate [latex]1[/latex]. Examples.

) is a parabola (as shown at the right).

θ b = {\displaystyle (x_{m},y_{m})\,} The point [latex](0,c)[/latex] is the [latex]y[/latex] intercept of the parabola.

If the ordinate of the maximum point of the corresponding parabola

The quadratic function graph can be easily derived from the graph of \(x^2.\). If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. − Note that the coefficient on [latex]x^2[/latex] (the one we call [latex]a[/latex]) is [latex]1[/latex].

(The superscript can be extended to negative numbers, referring to the iteration of the inverse of 2 The coefficient a controls the degree of curvature of the graph; a larger magnitude of a gives the graph a more closed (sharply curved) appearance. {\displaystyle (1-2x_{0})^{2^{n}}} (a, b, and c can have any value, except that a can't be 0.). a You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. 2 The graph of [latex]y=2x^2-4x+4. D

if the inverse exists.) {\displaystyle {\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,\,} + where A, B, C, D, and E are fixed coefficients and F is the constant term. resulting in, so again the vertex point coordinates, (h, k), can be expressed as, The roots (or zeros), r1 and r2, of the univariate quadratic function, When the coefficients a, b, and c, are real or complex, the roots are, The modulus of the roots of a quadratic A = ) Original figure by Mark Woodard. 1 + Recall that the quadratic equation sets the quadratic expression equal to zero instead of [latex]f(x)[/latex]: Now the quadratic formula can be applied to find the [latex]x[/latex]-values for which this statement is true. ) Since . θ {\displaystyle \theta } Smaller values of aexpand it outwards 3. max c To convert the standard form to vertex form, one needs a process called completing the square.

2 where x is the variable, and a, b, and c represent the coefficients. The graph of a quadratic function is a U-shaped curve called a parabola.

. The process involves a technique called completing the square. The vertex of a parabola is the place where it turns; hence, it is also called the turning point. 2 Firstly, we know h and k (at the vertex): So let's put that into this form of the equation: And so here is the resulting Quadratic Equation: Note: This may not be the correct equation for the data, but it’s a good model and the best we can come up with. The coefficients [latex]b[/latex] and [latex]a[/latex] together control the axis of symmetry of the parabola and the [latex]x[/latex]-coordinate of the vertex. When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". 2 Parabolas also have an axis of symmetry, which is parallel to the y-axis. This shape is shown below. x , For the given equation, we have the following coefficients: [latex]a = 1[/latex], [latex]b = -1[/latex], and [latex]c = -2[/latex]. ϕ − Note that half of [latex]6[/latex] is [latex]3[/latex] and [latex]3^2=9[/latex].

0 A quadratic function is a polynomial function of the form [latex]y=ax^2+bx+c[/latex]. The bivariate case in terms of variables x and y has the form. The solution of the logistic map when r=2 is, x

2 Recall that the [latex]x[/latex]-intercepts of a parabola indicate the roots, or zeros, of the quadratic function. 0 + > The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization, completing the square, graphing, Newton's method, or through the use of the quadratic formula. ⁡ Graph of \(x^2\) is basically the graph of the parent function of quadratic functions.. A quadratic function is a polynomial and their degree 2 which can be written in the general form, Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative: x is a root of f '(x) if f '(x) = 0 0 x In graphs of quadratic functions, the sign on the coefficient [latex]a[/latex] affects whether the graph opens up or down. B then the equation 1

y 5 [ 2 n where What if we have a graph, and want to find an equation? 2 ): We also know: the vertex is (3,−2), and the axis is x=3. > m {\displaystyle 4AB-E^{2}=0\,} The number of [latex]x[/latex]-intercepts varies depending upon the location of the graph (see the diagram below). You have already seen the standard form: Another common form is called vertex form, because when a quadratic is written in this form, it is very easy to tell where its vertex is located. There may be zero, one, or two [latex]x[/latex]-intercepts.

f [/latex] The black curve appears thinner because its coefficient [latex]a[/latex] is bigger than that of the blue curve. m 0 0 {\displaystyle x_{n}} Quadratics either open upward or downward: The blue parabola is the graph of [latex]y=3x^2. D So now we can plot the graph (with real understanding! D If there were, the curve would not be a function, as there would be two [latex]y[/latex] values for one [latex]x[/latex] value, at zero.

The coefficient c controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the y-axis. [/latex] The black parabola is the graph of [latex]y=-3x^2. A + where x and y are the variables and a, b, c, d, e, and f are the coefficients. + Parabola : The graph of a quadratic function is a parabola. is the golden ratio 2 +



Quadratic equations may take various forms. example. Graph of \(x^2\). ± Licensed CC BY-SA 4.0. π = Explain the meanings of the constants [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] for a quadratic equation in standard form. can be obtained, where

{\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}

We can still use the technique, but must be careful to first factor out the [latex]a[/latex] as in the following example: Consider [latex]y=2x^2+12x+5. < }, A bivariate quadratic function is a second-degree polynomial of the form.

| {\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}} In either case, the vertex is a turning point on the graph. : The black curve is [latex]y=4x^2[/latex] while the blue curve is [latex]y=3x^2. . First, identify the values for the coefficients: [latex]a = 1[/latex], [latex]b = - 4[/latex], and [latex]c = 5[/latex].

And negative values of aflip it upside down

x Substituting these into the quadratic formula, we have: [latex]x=\dfrac{-(-4) \pm \sqrt {(-4)^2-4(1)(5)}}{2(1)}[/latex], [latex]x=\dfrac{4 \pm \sqrt {16-20}}{2} \\ x=\dfrac{4 \pm \sqrt {-4}}{2}[/latex]. But there are some analytically tractable cases.

( 0 Therefore, it has no real roots. a where the initial condition parameter never repeats itself – it is non-periodic and exhibits sensitive dependence on initial conditions, so it is said to be chaotic. {\displaystyle 4AB-E^{2}<0\,}

) x

{\displaystyle f(x)} If f We can verify this algebraically. x 2 The roots of a quadratic function can be found algebraically with the quadratic formula, and graphically by making observations about its parabola. {\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}} {\displaystyle f^{(n)}(x)}

+ z

+ .

[/latex]: The axis of symmetry is a vertical line parallel to the y-axis at  [latex]x=1[/latex]. {\displaystyle x_{n}={\frac {1}{2}}-{\frac {1}{2}}(1-2x_{0})^{2^{n}}}, for x ) b a b a x , [/latex] We factor out the coefficient [latex]2[/latex] from the first two terms, writing this as: We then complete the square within the parentheses. = [latex]\displaystyle f(x)=ax^{2}+bx+c[/latex]. 0

The vertex form is given by: The vertex is [latex](h,k). ( x y ( A b If the coefficient [latex]a>0[/latex], the parabola opens upward, and if the coefficient [latex]a<0[/latex], the parabola opens downward. So we add and subtract [latex]9[/latex] within the parentheses, obtaining: We can then finish the calculation as follows: [latex]\begin{align} y&=2((x+3)^2-9)+5 \\ &=2(x+3)^2-18+5 \\ &=(x+3)^2-13 \end{align}[/latex], So the vertex of this parabola is [latex](-3,-13).[/latex]. y 0 We then both add and subtract this number as follows: Note that we both added and subtracted 4, so we didn’t actually change our function. n

− 1 | vertex: The maximum or minimum of a quadratic function.

+ is given by Possible [latex]x[/latex]-intercepts: A parabola can have no x-intercepts, one x-intercept, or two x-intercepts. x This … , (

Notice that we have [latex]\sqrt{-4}[/latex] in the formula, which is not a real number. )

) {\displaystyle g^{(n)}(x)} x Sometimes the word "order" is used with the meaning of "degree", e.g.



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