Grade 9

Intermediate

6 min

Lesson 3 of 12

Introduction to Quadratic Functions

Not Started

Explore the unique 'U' shape of parabolas and identify the key features of quadratic equations in standard form.

Ever wondered why a basketball always follows the same graceful arc toward the hoop, or why a flashlight beam forms a specific shape on a wall? That perfect 'U' shape isn't an accident—it's the power of the quadratic function in action.

Learning Objectives

1.Identify a quadratic function from its standard form equation $y = ax^2 + bx + c$ .
2.Locate the vertex, axis of symmetry, and y-intercept on a parabola.
3.Predict if a parabola opens upward or downward using the leading coefficient.

What Makes it Quadratic?

A quadratic function is any function that can be written in the standard form:

y = ax^2 + bx + c

The defining feature is the

x^2

term. For a function to be quadratic, the leading coefficient

a

must not be zero. If

a=0

, the

x^2

disappears, and you're left with a straight line! When graphed, these functions create a smooth, symmetrical curve called a parabola. Unlike linear functions that go in one direction forever, quadratics always have a 'turning point' where the graph changes direction.

Identify the values of $a$ , $b$ , and $c$ in the quadratic function: $y = 3x^2 - 5x + 2$ .

1. Look for the number in front of $x^2$ . Here, $a = 3$ . 2. Look for the number in front of $x$ . Remember to include the sign! Here, $b = -5$ . 3. Look for the constant term at the end. Here, $c = 2$ .

Quick Check

Is the function $y = 4x + 9$ a quadratic function? Why or why not?

Check the highest power of

x

in the equation.

Answer

No, because it does not have an $x^2$ term (the leading coefficient $a$ is 0).

The Anatomy of a Parabola

Every parabola has three 'landmarks' you need to know. First is the vertex: the highest or lowest point on the curve. Second is the axis of symmetry: an imaginary vertical line that passes through the vertex, cutting the parabola into two mirror-image halves. Its equation is always $x = h$ , where $h$ is the x-coordinate of the vertex. Finally, the y-intercept is the point where the curve crosses the y-axis. In standard form, the y-intercept is always at $(0, c)$ .

Find the y-intercept of the function $y = -2x^2 + 8x - 7$ .

1. Recall that the y-intercept occurs when $x = 0$ . 2. Plug $0$ into the equation: $y = -2(0)^2 + 8(0) - 7$ . 3. Simplify: $y = 0 + 0 - 7$ . 4. The y-intercept is $(0, -7)$ .

Quick Check

If a parabola's vertex is at $(4, -2)$ , what is the equation for its axis of symmetry?

The axis of symmetry is a vertical line passing through the x-coordinate of the vertex.

Answer

The axis of symmetry is $x = 4$ .

Up or Down? The Power of 'a'

You can predict the entire shape of a parabola just by looking at the sign of the leading coefficient $a$ . If $a$ is positive ( $a > 0$ ), the parabola opens upward, like a smiley face. This means the vertex is a minimum (the lowest point). If $a$ is negative ( $a < 0$ ), the parabola opens downward, like a frown. In this case, the vertex is a maximum (the highest point). The larger the absolute value of $a$ , the 'skinnier' the parabola becomes!

A ball is thrown in the air, following the path $h(t) = -16t^2 + 20t + 5$ , where $h$ is height and $t$ is time.

1. Identify $a$ : The coefficient of $t^2$ is $-16$ . 2. Determine direction: Since $-16$ is negative, the parabola opens downward. 3. Interpret: Because it opens downward, the vertex represents the maximum height the ball will reach.

Key Takeaways

1Quadratic functions follow the standard form $y = ax^2 + bx + c$ .
2The graph of a quadratic is a symmetrical 'U' shape called a parabola.
3The vertex is the turning point, and the axis of symmetry splits the parabola in half.
4If $a > 0$ , the parabola opens up; if $a < 0$ , it opens down.

Quick Check

Multiple Choice

Which of the following is a quadratic function in standard form?

Multiple Choice

If $a = -3$ in a quadratic equation, which way does the parabola open?

True/False

The y-intercept of $y = ax^2 + bx + c$ is always located at the point $(0, c)$ .

What's Next?

Review Tomorrow

In 24 hours, try to sketch a parabola that opens downward and label its vertex and axis of symmetry without looking at your notes.

Practice Activity

Find three quadratic equations in your textbook and identify the $a$ , $b$ , and $c$ values for each, then predict if they open up or down.

Browse

Browse

Introduction to Quadratic Functions

Learning Objectives

What Makes it Quadratic?

The Anatomy of a Parabola

Up or Down? The Power of 'a'

Key Takeaways

Quick Check

What's Next?

Introduction to Quadratic Functions

Learning Objectives

What Makes it Quadratic?

The Anatomy of a Parabola

Up or Down? The Power of 'a'

Key Takeaways

Quick Check

What's Next?