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$f(x)=a{x}^{2}+bx+c$

• 𝑎, 𝑏, and 𝑐 are constants,

• 𝑎≠0,

• 𝑥 is the variable.

The graph of a quadratic function is called a parabola.

It opens upwards if 𝑎>0 and downwards if 𝑎<0.

2.

The vertex of the parabola is the highest or lowest point, depending on whether it opens downwards or upwards, respectively.

The vertex form of a quadratic function is:

$f(x)=a(x-h{)}^{2}+k$

The vertex can also be found using the formula:

$h=-\frac{b}{2a}$

$$k=f(-\frac{b}{2a})$$

The axis of symmetry is the vertical line that passes through the vertex.

It has the equation:

x=ℎ

where ℎ is the x-coordinate of the vertex.

The roots of the quadratic function are the values of 𝑥 that make 𝑓(𝑥)=0

They can be found using the quadratic formula:

$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$

The discriminant determines the nature of the roots.

if

if

if

Transformations can be used to modify the graph of a quadratic function. Common transformations include:

Adding or subtracting a constant 𝑘 shifts the graph up or down.

$f(x)=a{x}^{2}+bx+(c+k)$

Adding or subtracting a constant ℎ to 𝑥 shifts the graph left or right.

$f(x)=a(x-h{)}^{2}+k$

Reflecting across the x-axis changes the sign of 𝑎.

$f(x)=-a{x}^{2}-bx-c$

Multiplying by a constant 𝑎 changes the width of the parabola.

if |a| > 1, the graph is stretched.

if 0 < |a| < 1, the graph is compressed.

Quadratic equations can be solved using several methods:

Write the quadratic equation in factored form:

$a{x}^{2}+bx+c=(dx+e)(fx+g)=0$

Rewrite the quadratic equation in vertex form by completing the square. Solve for 𝑥.

Use the quadratic formula:

$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$

Quadratic functions have numerous applications, including:

The path of a projectile under gravity is modeled by a quadratic function.

Example:

Quadratic functions are used in optimization to find maximum or minimum values.

Example: Maximizing the area of a rectangle given a fixed perimeter.

Quadratic functions model various economic phenomena such as profit, cost, and revenue functions.

Consider a quadratic function given by the equation $f(x)=a{x}^{2}+bx+c$ . You are told that the vertex of the parabola represented by this equation occurs at (-3,4) and it passes through the point (2,-1). Determine the values of a,b, and c.

Use vertex form of a quadratic equation.

A quadratic equation in vertex form is given by:

$f(x)=a(x-h{)}^{2}+k$

$$

We know that

$$

$$

Now that we know $a=-\frac{1}{5}$, substitute 𝑎 back into the vertex form and expand it to standard form:

$$

$$

Therefore, the values of 𝑎, 𝑏, and 𝑐 in the standard form $a{x}^{2}+bx+c$ of the given quadratic function are:

Consider a quadratic function defined by the equation $f(x)=a{x}^{2}+bx+c$ . The vertex of the parabola described by this equation is located at (1,-5) and it intersects the

Use vertex form of a quadratic equation.

The vertex form of a quadratic function is:

$f(x)=a(x-h{)}^{2}+k$

$$

Since the parabola intersects the y-axis at (0,3). we know

$$

Now that we have

$f(x)=8(x-1{)}^{2}-5$

Expanding this to standard form:

$$

Thus, the coefficients 𝑎a, 𝑏b, and 𝑐c in the standard form $a{x}^{2}+bx+c$ of the given quadratic function are:

$a=8,b=-16,c=3$

and the

You are given a quadratic function described by the equation

$f(x)=a{x}^{2}+bx+c$. This parabola has its vertex at (2,-7) and it crosses the x-axis at two points, one of which is (5,0). Determine the values of

Use the vertex form of a quadratic equation.

The vertex form of a quadratic function is:

$f(x)=a(x-h{)}^{2}+k$

$f(x)=a(x-2{)}^{2}-7$

Since the parabola touches the

Now we have $a=\frac{7}{9}$, so substitute it back:

Thus, the coefficients 𝑎, 𝑏, and 𝑐 in the standard form $a{x}^{2}+bx+c$ of the given quadratic function are:

$f(x)=\frac{7}{9}{x}^{2}-\frac{28}{9}x-\frac{35}{9}$.

It visually illustrates the vertex at (2,-7) and the

This graph helps in understanding the properties of the parabola, especially how it interacts with the

You are given a quadratic function described by the equation $f(x)=a{x}^{2}+bx+c$.

The vertex of the parabola is located at (-2,3), and it crosses the x-axis at points (-5,0) and (1,0). Determine the values of

Use the vertex form of a quadratic equation.

The vertex form of a quadratic function is:

$f(x)=a(x-h{)}^{2}+k$

where (

$f(x)=a(x+2{)}^{2}+3$

Given the

Substitute -5 for

For

Given the

Substitute -5 for

For

Now we have $a=\frac{-1}{2}$, substitute back into the vertex form and expand:

Thus, the coefficients a,b, and c in the standard form $a{x}^{2}+bx+c$ of the given quadratic function are:

Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

Quadratic functions, typically expressed in the form y=ax^2+bx+c, have numerous practical applications in various fields, making them an essential concept in algebra. Here are some key applications:

Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

**Projectile Motion:**Quadratic functions describe the path of a projectile under the influence of gravity. The equation y=ax^2+bx+c represents the trajectory of an object, such as a ball being thrown or a rocket being launched.Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

**Structural Engineering:**Quadratic functions are used to model the shapes of parabolic arches and bridges, optimizing material use and ensuring structural stability.Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: The St. Louis Gateway Arch, which has a parabolic shape, can be modeled and analyzed using quadratic functions.
Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: The St. Louis Gateway Arch, which has a parabolic shape, can be modeled and analyzed using quadratic functions.

**Revenue and Profit Maximization:**Quadratic functions are used to model revenue and profit functions. The vertex of the parabola indicates the maximum or minimum point, which is crucial for determining optimal pricing strategies.Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: If the revenue $R$ from selling $x$ units of a product is given by R(x)=ax^2+bx+c, finding the vertex helps in determining the quantity that maximizes revenue.
Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: If the revenue $R$ from selling $x$ units of a product is given by R(x)=ax^2+bx+c, finding the vertex helps in determining the quantity that maximizes revenue.

**Population Growth:**Quadratic functions can model the growth of populations under certain conditions, such as limited resources, where the growth rate eventually decreases as the population size approaches the carrying capacity.Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: The logistic growth model in its simplest form can be approximated by a quadratic function in its early stages.
Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: The logistic growth model in its simplest form can be approximated by a quadratic function in its early stages.

**Portfolio Optimization:**Quadratic functions are used in mean-variance optimization to determine the portfolio with the maximum expected return for a given level of risk.Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: The quadratic function in the context of finance can represent the trade-off between risk (variance) and return in an investment portfolio.
Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

- Example: The quadratic function in the context of finance can represent the trade-off between risk (variance) and return in an investment portfolio.

**Bezier Curves:**Quadratic Bezier curves are used in computer graphics and animation to create smooth curves. These curves are defined by a quadratic function and are essential for designing scalable vector graphics and animations.Quadratic Functions in Algebra (IB Mathematics Analysis and Approaches Standard Level)

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