Quadratic Functions

Welcome, students! In this lesson, you will explore one of the most important function types in mathematics: the quadratic function 📈. Quadratic functions show up in many real situations, from the path of a basketball 🏀 to the shape of a bridge cable, from business profit models to the area of a garden. By the end of this lesson, you should be able to explain the key ideas and vocabulary, interpret graphs, connect the algebra to the shape, and use technology to analyze quadratic relationships.

Learning objectives:

Explain the main ideas and terminology behind quadratic functions.
Apply IB Mathematics: Applications and Interpretation SL reasoning and procedures related to quadratic functions.
Connect quadratic functions to the broader topic of functions.
Summarize how quadratic functions fit within functions.
Use evidence and examples related to quadratic functions in context.

A quadratic function is a function that can be written in the form $f(x)=ax^2+bx+c$ where $a\neq 0$. The highest power of $x$ is $2$, so this is a polynomial function of degree $2$. Its graph is called a parabola, and that curve is one of the most recognizable shapes in mathematics.

What Makes a Quadratic Function Special?

A quadratic function is different from a linear function because its rate of change is not constant. For a linear function, the graph is a straight line and the change in $y$ for each equal change in $x$ stays the same. For a quadratic function, the graph bends, so the change in slope keeps changing.

The general form $f(x)=ax^2+bx+c$ tells us important information:

$a$ controls the opening direction and the width of the parabola.
$b$ helps position the graph and affects the axis of symmetry.
$c$ is the $y$-intercept, because $f(0)=c$.

If $a>0$, the parabola opens upward and has a minimum point. If $a<0$, the parabola opens downward and has a maximum point. This is useful in real-life modeling. For example, if a company is modeling profit, the graph might open downward because profit rises at first and then falls after too much production or too much price increase.

The turning point of the parabola is called the vertex. The vertex is the highest or lowest point on the graph, depending on whether the parabola opens down or up. The line passing through the vertex and splitting the graph into two equal halves is called the axis of symmetry.

For students, this means a quadratic function is not just an algebraic rule. It is a pattern of change that often describes situations where something increases, reaches a peak, and then decreases, or decreases, reaches a minimum, and then increases again.

Forms of Quadratic Functions and What They Show

Quadratic functions can be written in several forms, and each form is useful for different purposes.

Standard form

The standard form is $f(x)=ax^2+bx+c$. This form is especially useful for identifying the $y$-intercept and using algebraic methods to solve equations.

Vertex form

The vertex form is $f(x)=a(x-h)^2+k$. In this form, the vertex is $(h,k)$, and the axis of symmetry is $x=h$. This form is excellent when you want to describe the graph by its turning point.

For example, the function $f(x)=2(x-3)^2-5$ has vertex $(3,-5)$, opens upward because $2>0$, and is narrower than the basic parabola because the value of $|a|$ is greater than 1`.

Factored form

The factored form is $f(x)=a(x-r_1)(x-r_2)$, where $r_1$ and $r_2$ are the roots or zeros. These are the $x$-values where the graph crosses the $x$-axis, so they satisfy $f(x)=0$.

This form is useful when solving quadratic equations by factoring. If you know the zeros, you can describe where the graph touches or crosses the horizontal axis.

A key skill in IB Mathematics: Applications and Interpretation SL is moving between these forms depending on the context. A graphing calculator or technology can help check your work, but you still need to understand what each form means.

Graphing Quadratic Functions and Interpreting Features

The graph of a quadratic function is a parabola. Its most important features are the vertex, axis of symmetry, intercepts, and opening direction.

Suppose $f(x)=x^2-4x+3$. To find the $y$-intercept, set $x=0$:

$$f(0)=3$$

So the $y$-intercept is $(0,3)$.

To find the zeros, solve $x^2-4x+3=0$. Factoring gives

$$f(x)=(x-1)(x-3)$$

So the zeros are $x=1$ and $x=3$.

The vertex can be found by completing the square or using the formula $x=-\frac{b}{2a}$. Here, $a=1$ and $b=-4$, so

$$x=-\frac{-4}{2(1)}=2$$

Then

$$f(2)=2^2-4(2)+3=-1$$

So the vertex is $(2,-1)$, and the axis of symmetry is $x=2$.

This graph crosses the $x$-axis at two points, so in context it could represent a situation with two solutions. For example, if $f(x)$ represented the height of a ball relative to the ground, the zeros could show when the ball is on the ground.

When interpreting graphs, always connect the math to the story. If $x$ is time, then the vertex might represent the maximum height, minimum cost, or best value achieved during the process.

Quadratic Functions in Context and Technology

Quadratic functions are often used in functional models in context. In real life, a quadratic model is useful when the relationship is curved and has a turning point.

For example, imagine a student designing a rectangular garden with a fixed amount of fencing. The area might increase as the dimensions change, but after a certain point the area begins to decrease. This creates a quadratic relationship.

Another common example is projectile motion. The height of a thrown object often follows a quadratic pattern because gravity acts in a constant downward way. If the height is modeled by $h(t)=at^2+bt+c$, then the maximum height occurs at the vertex if $a<0$.

Technology is very important in IB Mathematics: Applications and Interpretation SL. A graphing calculator can help you:

plot the parabola,
find the vertex,
estimate roots,
compare data with a model,
and decide whether a quadratic model fits the situation well.

Suppose a set of data points suggests a curved pattern. You can use quadratic regression to find a model of the form $y=ax^2+bx+c$ that best fits the data. Regression does not mean the model is perfect; it means the model is chosen to minimize the overall error between the data and the curve.

If you are analyzing a scatter plot, ask:

Does the pattern rise and then fall, or fall and then rise?
Is there a clear turning point?
Does a quadratic curve match the shape better than a line?

These questions help students decide when a quadratic model is appropriate.

Solving Quadratic Problems and Making Decisions

Quadratic functions are often used to solve practical problems, not just algebra exercises. The main tasks are finding roots, finding the vertex, and interpreting results.

If a quadratic equation is given by $ax^2+bx+c=0$, then the solutions can be found by factoring, completing the square, or using the quadratic formula:

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

The expression under the square root, $b^2-4ac$, is called the discriminant. It tells us how many real solutions there are:

If $b^2-4ac>0$, there are two real solutions.
If $b^2-4ac=0$, there is one real solution.
If $b^2-4ac<0$, there are no real solutions.

This is useful in context. For example, if a projectile’s height equation has no real zeros, it may mean the object never hits the ground in the model’s time interval.

Quadratic functions also help with optimization. If you want to find the maximum area, maximum profit, or minimum cost, the vertex gives the best value.

For instance, if profit is modeled by $P(x)=-2x^2+20x-18$, then the maximum profit occurs at

$$x=-\frac{b}{2a}=-\frac{20}{2(-2)}=5$$

Then

$$P(5)=-2(25)+20(5)-18=32$$

So the maximum profit is $32$ when $x=5$. In context, this could mean producing 5 units, choosing a price level of 5, or using 5 hours of effort depending on the situation.

When using quadratic functions in context, always check whether the solution is reasonable. A negative time or a decimal number of people would not make sense in many real situations.

Conclusion

Quadratic functions are a major part of the study of functions because they show how a rule can create a curved graph with a clear turning point. In IB Mathematics: Applications and Interpretation SL, students needs to understand the algebra, the graph, and the context together. The standard form, vertex form, and factored form each reveal different information. Graphs help us interpret roots, intercepts, and maximum or minimum values. Technology supports analysis, especially for regression and checking models. Quadratic functions are powerful because they describe real patterns where values rise and fall in a balanced, predictable way.

Study Notes

A quadratic function has the form $f(x)=ax^2+bx+c$, where $a\neq 0$.
Its graph is a parabola.
If $a>0$, the parabola opens upward and has a minimum vertex.
If $a<0$, the parabola opens downward and has a maximum vertex.
The $y$-intercept is $c$, because $f(0)=c$.
The axis of symmetry is the vertical line through the vertex.
The vertex form is $f(x)=a(x-h)^2+k$, with vertex $(h,k)$.
The factored form is useful for finding zeros, where $f(x)=0$.
The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
The discriminant $b^2-4ac$ tells how many real solutions there are.
Quadratic regression helps fit a parabola to data when the relationship curves.
In context, quadratic models are useful for height, area, profit, and other situations with a turning point.
Always interpret solutions carefully so they make sense in the real world.