The Main Idea 

Quadratic functions graph into a curve called a parabola. The direction in which the parabola opens is determined by the sign of the coefficient [latex]a[/latex] in the quadratic function’s general form [latex]f(x)=ax^2+bx+c[/latex]. If [latex]a>0[/latex], the parabola opens upward like a smile, and if [latex]a<0[/latex], it opens downward like a frown.

There are two forms of equations quadratics can be represented in:

• General Form: [latex]f(x) = ax^2 + bx + c[/latex], where [latex]a[/latex], [latex]b[/latex]>, and [latex]c[/latex] are constants and [latex]a \neq 0[/latex].
• Standard Form: [latex]f(x) = a(x - h)^2 + k[/latex], where [latex](h, k)[/latex] is the vertex of the parabola.

Here are some key features of quadratic functions:

• Vertex: The turning point of the parabola, which can be a minimum or maximum.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror images, given by [latex]x = -\frac{b}{2a}[/latex].
• [latex]Y[/latex]-Intercept: The point where the parabola crosses the [latex]y[/latex]-axis, which is at [latex]f(0)[/latex].
• [latex]X[/latex]-Intercepts: Points where the parabola crosses the [latex]x[/latex]-axis, found by setting [latex]f(x) = 0[/latex].

Step-by-Step: Finding the Vertex

1. From the general form, identify [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex].
2. Calculate [latex]h[/latex] using [latex]h = -\frac{b}{2a}[/latex].
3. Determine [latex]k[/latex] by evaluating [latex]f(h)[/latex].

The Main Idea 

Quadratic functions form parabolas on a graph, and these parabolas have specific domains and ranges that dictate their possible values.

• Domain Details: The domain of a quadratic function includes all real numbers. There are no restrictions on the [latex]x[/latex]-values that can be input into the function.
• Range Rules: The range of a quadratic function depends on the direction the parabola opens. If the parabola opens upwards, the range is all [latex]y[/latex]-values greater than or equal to the vertex’s [latex]y[/latex]-value. If it opens downwards, the range is all [latex]y[/latex]-values less than or equal to the vertex’s [latex]y[/latex]-value.

The vertex can tell you a lot about the range of a quadratic function.

• Positive ‘a’ Value: For [latex]f(x) = ax^2 + bx + c[/latex] with a positive ‘[latex]a[/latex]‘, the parabola opens upwards. The vertex represents the minimum [latex]y[/latex]-value, so the range is [latex]y \geq f(-\frac{b}{2a})[/latex].
• Negative ‘a’ Value: For [latex]f(x) = ax^2 + bx + c[/latex] with a negative ‘[latex]a[/latex]‘, the parabola opens downwards. The vertex represents the maximum [latex]y[/latex]-value, so the range is [latex]y \leq f(-\frac{b}{2a})[/latex].

The Main Idea 

A parabola is a U-shaped curve that reflects the behavior of objects under the influence of gravity or the shape of satellite dishes capturing signals. It’s a visual representation of a quadratic function, showcasing how variables interact to create a curve that opens upwards or downwards.

• Vertex: The vertex is the peak or the lowest point of the parabola, serving as a critical point that determines the maximum or minimum value of the quadratic function.
• Axis of Symmetry: This is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It’s a guide to finding the vertex and understanding the parabola’s symmetry.
• Zeros: These are the points where the parabola intersects the x-axis, also known as solutions or roots of the quadratic equation, where [latex]y=0[/latex].
• [latex]Y[/latex]-Intercept: The point where the parabola crosses the y-axis, indicating the value of [latex]y[/latex] when [latex]x=0[/latex].

The Main Idea 

Creating a graph of a function is one way to understand the relationship between the inputs and outputs of that function. Creating a graph can be done by choosing values for [latex]x[/latex], finding the corresponding [latex]y[/latex] values, and plotting them. However, it helps to understand the basic shape of the function. Knowing how changes to the basic function equation affect the graph is also helpful.

The shape of a quadratic function is a parabola. Parabolas have the equation [latex]f(x)=ax^{2}+bx+c[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers and [latex]a\ne0[/latex]. The value of [latex]a[/latex] determines the width and the direction of the parabola, while the vertex depends on the values of [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]. The vertex is [latex]\displaystyle \left( \dfrac{-b}{2a},f\left( \dfrac{-b}{2a} \right) \right)[/latex].

In the following video, we show an example of plotting a quadratic function using a table of values.

The Main Idea 

The vertex of a parabola is like the peak of a mountain or the bottom of a valley, depending on which way it opens. It’s the most defining feature, determining the lowest or highest point on the graph. In the equation [latex]f(x) = a(x-h)^2 + k[/latex], the coordinates [latex](h, k)[/latex] pinpoint the vertex’s location on the Cartesian plane. Shifting the Graph

• Vertical Shifts: The [latex]k[/latex] value dictates the vertical shift. If [latex]k > 0[/latex], the graph moves upward; if [latex]k < 0[/latex], it moves downward. Think of it as lifting or pressing down on the curve.
• Horizontal Shifts: The [latex]h[/latex] value controls the horizontal shift. A positive [latex]h[/latex] shifts the graph to the right, while a negative [latex]h[/latex] shifts it to the left. The direction is opposite to the sign inside the parentheses due to the equation’s structure.

Stretch or Compress the Graph

• Vertical Stretch/Compression: The coefficient [latex]a[/latex] affects the parabola’s width. If [latex]|a| > 1[/latex], the graph becomes narrower; if [latex]|a| < 1[/latex], it becomes wider. It’s like zooming in and out on the curve vertically.

Quick Tips: Applying Transformations

• To vertically shift the graph, modify the [latex]k[/latex] value accordingly.
• To horizontally shift the graph, change the [latex]h[/latex] value, keeping in mind the inverse relationship with the direction of the shift.
• To alter the width of the graph, adjust the [latex]a[/latex] value, which stretches or compresses the parabola.