Quadratic Functions 📈

Welcome, students. In this lesson, you will explore quadratic functions, one of the most important families of functions in mathematics and in real-world modeling. Quadratic functions appear whenever a quantity changes in a curved way, such as the path of a thrown ball, the shape of a suspension bridge cable, or the profit from a business decision. By the end of this lesson, you should be able to explain what makes a function quadratic, identify key features on its graph, and interpret quadratic models in context.

What is a quadratic function?

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 the graph is a parabola. The value of $a$ controls whether the parabola opens upward or downward. If $a>0$, the graph opens upward like a smile 😀. If $a<0$, it opens downward like a frown.

The number $c$ is the $y$-intercept, because $f(0)=c$. This means the graph crosses the $y$-axis at $(0,c)$. The coefficients $a$, $b$, and $c$ each affect the shape and position of the graph, so quadratic functions are useful for analyzing relationships where change is not constant.

In the broader study of functions, quadratic functions matter because they show how algebra, graphs, and context work together. A function gives an input-output relationship, and a quadratic function is a special type of function with a curved graph. In IB Mathematics: Applications and Interpretation HL, this connects to modeling, interpretation, and technology-based analysis.

Forms of quadratic functions and what they tell us

Quadratic functions can be written in different forms, and each form highlights different information.

The standard form is $f(x)=ax^2+bx+c$. This form is useful for identifying the coefficients directly and finding the $y$-intercept.

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 especially helpful when analyzing transformations. For example, the graph of $f(x)=(x-3)^2-4$ is the graph of $y=x^2$ shifted right $3$ units and down $4$ units.

The factored form is $f(x)=a(x-r_1)(x-r_2)$, where $r_1$ and $r_2$ are the zeros or roots. These are the $x$-values where $f(x)=0$. If the quadratic has only one repeated root, then the graph just touches the $x$-axis at that point.

For example, consider $f(x)=2(x+1)(x-5)$. The zeros are $x=-1$ and $x=5$, so the graph crosses the $x$-axis at $(-1,0)$ and $(5,0)$. Since $a=2>0$, the parabola opens upward.

Understanding these forms helps you move between algebraic expressions and graphical meaning, which is a major goal in this topic.

Key features of the graph

Every quadratic graph has a vertex, an axis of symmetry, and a parabola shape. The vertex is the turning point of the graph. If $a>0$, the vertex is the minimum point. If $a<0$, the vertex is the maximum point.

The axis of symmetry is a vertical line that splits the parabola into two matching halves. If a quadratic is written in standard form, the axis of symmetry is $x=-\frac{b}{2a}$. This result is very useful because it gives the $x$-coordinate of the vertex.

For example, for $f(x)=x^2-6x+5$, we have $a=1$ and $b=-6$, so the axis of symmetry is $x=-\frac{-6}{2(1)}=3$. To find the vertex, evaluate the function at $x=3$: $f(3)=3^2-6(3)+5=9-18+5=-4$. So the vertex is $(3,-4)$.

The roots of a quadratic can be found by factoring, using the quadratic formula, or completing the square. The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. This formula works for every quadratic function written as $ax^2+bx+c=0$.

The discriminant is $b^2-4ac$. It tells you how many real roots a quadratic has:

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

This is useful in interpretation. For example, if a quadratic model represents the height of a ball above the ground, real roots can represent the times when the ball is on the ground.

Quadratic transformations and graph behavior

Quadratic graphs can be transformed by changing the equation. The parent function is $y=x^2$. From this graph, you can create many others using transformations.

If a function is $f(x)=a(x-h)^2+k$, then:

$h$ shifts the graph horizontally.
$k$ shifts the graph vertically.
The value of $a$ stretches or compresses the graph, and may reflect it across the $x$-axis.

For instance, $f(x)=-3(x+2)^2+1$ has vertex $(-2,1)$ and opens downward because $a=-3$. The graph is narrower than $y=x^2$ because $|a|=3>1$.

Transformations are important in technology-supported analysis. A graphing calculator or software can quickly show how changing $a$, $h$, or $k$ changes the parabola. This helps you test patterns, compare models, and match a function to data.

Real-life example: suppose a company models profit as $P(x)=-2x^2+40x-150$, where $x$ is the number of items sold in hundreds. The negative coefficient means profit eventually decreases after a certain point. The vertex gives the maximum profit, which can help decide the best production level.

Solving quadratic equations in context

Quadratic functions often appear when you need to solve a problem involving a maximum, minimum, or crossing point. If a question asks when a quantity becomes zero, you are usually solving a quadratic equation.

Example: A ball is launched, and its height in meters after $t$ seconds is given by $h(t)=-5t^2+20t+1$. To find when the ball hits the ground, solve $h(t)=0$:

$$-5t^2+20t+1=0$$

You could use the quadratic formula with $a=-5$, $b=20$, and $c=1$:

$$t=\frac{-20\pm\sqrt{20^2-4(-5)(1)}}{2(-5)}$$

$$t=\frac{-20\pm\sqrt{420}}{-10}$$

This gives two solutions, but only the positive one makes sense in context because time cannot be negative here. That is a key IB skill: always check whether the answer fits the situation.

Another common task is finding the maximum height. Since the graph opens downward, the vertex gives the maximum. The time of maximum height is $t=-\frac{b}{2a}=2$. Then $h(2)=-5(2)^2+20(2)+1=21$. So the maximum height is $21$ meters.

This is a powerful example of how algebra and real-world meaning work together. The numbers are not just symbols; they describe motion, change, and limits in the real world.

Regression and fitting quadratic models

In statistics and modeling, quadratic functions can be fitted to data when the relationship is curved rather than linear. This is common in situations where values rise and then fall, or fall and then rise.

Suppose a set of data points shows the height of a fountain spray at different horizontal distances. A quadratic regression model can estimate the curve that best fits the data. Technology is especially helpful here because it can calculate the regression equation quickly and display the graph.

When you use quadratic regression, it is important to interpret the model carefully:

The equation describes the pattern in the data.
The model may be useful only within the range of the observed data.
Predictions far outside the data range may be unreliable.

For example, if the data measures the distance of a thrown object from $x=0$ to $x=8$, using the model to predict behavior at $x=100$ is not sensible. This is called extrapolation, and it can lead to inaccurate conclusions.

In IB Mathematics: Applications and Interpretation HL, you should be ready to discuss how well a quadratic model fits the data, what the parameters mean, and whether the model makes sense in context. Good modeling is not just about getting an equation; it is about interpreting the equation responsibly.

Conclusion

Quadratic functions are a central part of the Functions topic because they connect algebraic structure, graph behavior, and real-world interpretation. You have seen how the form $ax^2+bx+c$ describes a parabola, how vertex and factored forms reveal useful information, and how roots, turning points, and transformations help you analyze problems. You have also seen how quadratic regression is used to fit data and how technology supports this process.

students, when you study quadratic functions, focus on meaning as well as calculation. Ask what the graph tells you, what the coefficients do, and whether the answer makes sense in context. That approach is exactly what IB Mathematics: Applications and Interpretation HL expects.

Study Notes

A quadratic function has the form $f(x)=ax^2+bx+c$, where $a\neq 0$.
The graph of a quadratic function is a parabola.
If $a>0$, the parabola opens upward; if $a<0$, it opens downward.
The vertex is the turning point of the parabola.
The axis of symmetry is $x=-\frac{b}{2a}$.
The $y$-intercept is $c$, because $f(0)=c$.
The roots are the solutions to $f(x)=0$.
The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
The discriminant $b^2-4ac$ tells how many real roots there are.
Vertex form is $f(x)=a(x-h)^2+k$, which shows the vertex $(h,k)$ directly.
Factored form is $f(x)=a(x-r_1)(x-r_2)$, which shows the roots directly.
Quadratic models are useful for motion, profit, area, and other curved relationships.
Technology helps with graphing, regression, and checking models against data.
Always interpret answers in context and check whether they are realistic.