Definitions
Graphing
Solving Quadratics
Applications
Properties
100
What is a quadratic function?
A function of the form f(x) = ax^2 + bx + c
100
What is the shape of the graph of a quadratic function?
A parabola.
100
What method can be used to solve the equation ax^2 + bx + c = 0 $?
Factoring, completing the square, or using the quadratic formula.
100
In what real-world scenario might a quadratic function be used?
Projectile motion, such as the path of a thrown ball.
100
What is the axis of symmetry in a quadratic function?
Theverticallinethatdividestheparabolaintotwomirror
200
What is the standard form of a quadratic function?
Thestandardformis f(x) = ax^2 + bx + c
200
What is the vertex of a parabola?
The highest or lowest point of the parabola, depending on its orientation.
200
What is the quadratic formula?
200
How can you determine the maximum height of a projectile modeled by a quadratic function?
By finding the vertex of the parabola.
200
What does the discriminant
It indicates the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.
300
What is the vertex form of a quadratic function, and how can you identify the vertex from this form?
The vertex form of a quadratic function is f(x) = a(x - h)^2 + k ,where the vertex is the point (h, k) $.
300
What effect does changing the value of a in the vertex form f(x) = a(x - h)^2 + k have on the graph of the quadratic function?
Changing the value of a affects the width and direction of the parabola ;if a > 0, the parabola opens upwards, an dif a < 0 ,it opens downwards. Alarger absolute value of a $ makes the parabola narrower, while a smaller absolute value makes it wider.
300
If a quadratic function in vertex form is given as f(x) = \fraction{1}{2}(x - 1)^2 + 3 ,what is the effect of the coefficient \fraction{1}{2} on the graph
The coefficient \fraction{1}{2} makes the parabola wider than the standard parabola y = x^2 $ because it compresses the vertical stretch, resulting in a less steep graph.
300
What is the process called when you rewrite a quadratic function from standard form to vertex form?
The process is called "completing the square."
300
In the vertex form f(x) = a(x - h)^2 + k ,what does the value of k represent in relation to the graph
The value of k $ represents the vertical shift of the parabola; it indicates how far the graph is shifted up or down from the x-axis.
400
How can you convert a quadratic function from standard form f(x) = ax^2 + bx + c to vertex form f(x) = a(x - h)^2 + k ?
You can convert it by completing the square on the ax^2 + bx part of the equation to express it in the form a(x - h)^2 + k $.
400
If a quadratic function is given in vertex form f(x) = 2(x - 3)^2 + 5 ,what are the coordinates of the vertex?
The coordinates of the vertex are (3, 5) $.
400
How can you determine they−intercept of a quadratic function given in vertex form f(x) = a(x - h)^2 + k ?
To find they−intercept, set x = 0 in the vertex form equation and solve for f(0) $, which gives you the y-coordinate of the y-intercept.
400
What is the significance of the values h and k in the vertex form f(x) = a(x - h)^2 + k ?
The values h and k represent the coordinates of the vertex of the parabola, where (h, k) is the highest or lowest point of the graph, depending on the sign of a $.
400
In the vertex form f(x) = -3(x + 2)^2 + 4 $, what does the negative sign in front of the coefficient indicate about the graph of the function?
The negative sign indicates that the parabola opens downward, meaning the vertex is the maximum point of the graph.
500
Given the quadratic function in vertex form f(x) = -4(x - 2)^2 + 6 ,how would you describe the transformations applied to the parent function f(x) = x^2 ?
The function has been transformed by shifting 2 units to the right(dueto (x - 2) ),shifting 6 units up(dueto +6 ),and reflecting over the x−axis(duetothenegativecoefficient-4$), while also vertically stretching the graph by a factor of 4.
500
How can you derive the vertex form of a quadratic function from its standard form f(x) = ax^2 + bx + c using the method of completing the square?
To derive the vertex form, first factor out a from the ax^2 + bx terms, then complete the square by adding and subtracting \left(\frac{b}{2a}\right)^2 inside the parentheses. This results in the expression a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a} ,which can be rewritten as f(x) = a(x - h)^2 + k ,where h = -\frac{b}{2a} and k = f(h) $.
500
If a quadratic function is given in vertex form as f(x) = 3(x + 1)^2 - 5 ,how would you find the x−intercept so the function, and what are they?
To find the x−intercepts, set f(x) = 0 : $$ 0 = 3(x + 1)^2 - 5 $$ Add 5 to both sides: $$ 5 = 3(x + 1)^2 $$ Divide by 3: $$ \frac{5}{3} = (x + 1)^2 $$ Take the square root of both sides: $$ x + 1 = \pm \sqrt{\frac{5}{3}} $$ Subtract 1 from both sides: $$ x = -1 \pm \sqrt{\frac{5}{3}} $$ Thus, the x-intercepts are x = -1 + \sqrt{\frac{5}{3}} and x = -1 - \sqrt{\frac{5}{3}} $.
500
In the vertex form f(x) = -2(x - 4)^2 + 8 ,how can you determine the maximum value of the function, and what is that value?
The maximum value of the function occurs at the vertex, which is given by the coordinates (h, k) in the vertex form f(x) = a(x - h)^2 + k .Here, h = 4 and k = 8 .Since the parabola opens downward(because a = -2 < 0 ),the maximum value of the function is 8 $.
500
How can you determine the axis of symmetry for the quadratic function given in vertex form f(x) = \frac{1}{3}(x - 5)^2 + 2 ,and what is the equation of the axis of symmetry?
The axis of symmetry for a quadratic function in vertex form f(x) = a(x - h)^2 + k is given by the line x = h .In this case, h = 5 ,so the equation of the axis of symmetry is x = 5 $.