Graph Linear and Quadratic Functions

Deck 1 of 2: Lines and Standard Form Parabolas

In this deck:

• Graph linear functions from three different forms
• Graph quadratics from standard form using the five-step process

What You Will Learn Today

1. Graph linear functions from three forms
2. Graph quadratics from standard form — five key features
3. Graph quadratics from vertex and factored forms
4. Read roots from factored form
5. Identify quadratic maximum or minimum
6. Label intercepts, vertex, and axis of symmetry

Three Forms, Always the Same Line

Three forms — same line:

• Slope-intercept: — slope and y-intercept immediate
• Standard form: — both intercepts direct
• Point-slope: — point and slope given

Slope-Intercept Form: Finding Both Intercepts

• Slope:
• y-intercept: →
• x-intercept: → →

Plot both intercepts, connect with a line, confirm slope .

Standard Form: Intercept Method for Graphing

Find intercepts directly:

• x-intercept: set : → →
• y-intercept: set : → →

Plot and , draw the line.

Point-Slope Form: Plot Point, Then Find Intercepts

• Known point: ; slope
• Convert: ; y-int ; x-int

Linear Graphing: Key Features Summary

Every linear graph must show:

• Both intercepts labeled
• Slope confirmed visually

The form determines the fastest path — the final graph is always the same.

Check: Graph y = −3x + 9

Find both intercepts of and graph the line.

• y-intercept: plug in →
• x-intercept: set , solve for →

Write the intercept coordinates before the next slide.

Answer: Intercepts of y = −3x + 9

• y-intercept: →
• x-intercept: → →

Common mistake: Confusing which variable to set to zero.

Memory: x-intercept → y IS zero. y-intercept → x IS zero.

From Linear Functions to Parabolas

Linear: constant rate → straight line

Quadratic : varying rate → parabola

New features:

• Vertex — turning point
• Axis of symmetry — vertical line through vertex
• Direction — up () or down ()

Five-Step Process for Standard Form

Given :

1. y-intercept:
2. Axis of symmetry:
3. Vertex: evaluate at the axis value
4. x-intercepts: solve
5. Direction: (up, minimum) or (down, maximum)

Standard Form Example: Two Intercepts

1. y-int: · 2. Axis: · 3. Vertex:
2. x-ints: factor → and · 5. Opens up

When the Discriminant Is Negative: No Roots

Discriminant: → no real roots

• y-int: ; Axis: ; Vertex:
• Opens up; parabola sits entirely above the x-axis

Use the Discriminant Before You Solve

Maximum vs. Minimum: Sign of a

• : parabola opens upward (U-shape) → vertex is a minimum
• : parabola opens downward (∩-shape) → vertex is a maximum

The maximum/minimum VALUE is the y-coordinate of the vertex.

It occurs at the x-coordinate of the vertex.

Check: Identify Vertex, Intercepts, and Direction

For :

1. How many x-intercepts does this function have?
2. Where is the vertex?
3. Does this function have a maximum or minimum?

Answer all three before the next slide.

Answer: h(x) = x² + 4

1. Discriminant: → no x-intercepts
2. Vertex: axis at ; → vertex at
3. → opens upward → minimum value of 4

The graph is a U-shape sitting above the x-axis.

Key Takeaways from Deck One

✓ Three linear forms → same line; label both intercepts

✓ Standard form: 5 steps → y-int, axis, vertex, x-ints, direction

✓ = min; = max

x-intercept: set ; y-intercept: set

Max/min VALUE = y-coordinate of vertex

Coming Up in Deck Two

Vertex Form, Factored Form, and Max/Min in Context

• Read vertex directly from
• Read roots directly from
• Three-form comparison: which reveals what
• Maximum height, maximum revenue — real applications

Applies LOs 3, 4, 5, and 6