Function Formula Identification: Matching Graph 1's Quadratic Curve

Q: How can I tell if a graph shows a quadratic function?

Look for the U-shape or upside-down U! Quadratic functions always create parabolas - curved lines that bend. Linear functions make straight lines, while quadratic functions make curves.

Q: What does the vertex of this parabola tell me?

The vertex is the lowest point (or highest if opening down). For $ y = x^2 - 6x + 8 $, the vertex is at (3, -1), which matches the bottom of the curve in the graph.

Q: How do I find where the parabola crosses the x-axis?

Set y = 0 and solve: $ 0 = x^2 - 6x + 8 $. Factor to get $ (x-2)(x-4) = 0 $, so it crosses at x = 2 and x = 4.

Q: What if I can't see the exact coordinates clearly?

Focus on the overall shape first! If it's curved like a U, it must be quadratic. Then look for general features like whether it opens up or down, and approximate where key points might be.

Quadratic Functions with Graph Analysis

Choose the formula that describes graph 1:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Match the function to graph 1

00:03 The graph is a smiling parabola

00:08 We'll use the formula for a positive parabola

00:17 According to the formula, we'll eliminate the unsuitable options

00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the formula that describes graph 1:

Step-by-step solution

To solve this problem, we need to determine whether the provided graph corresponds to a quadratic or linear function.

First, we observe the shape of the graph.
The graph shows a downward curve, indicating it is a parabola.
The general form of a quadratic equation $(y = ax^2 + bx + c)$ implies graph types.
Let's verify if one of the given quadratic choices represents this graph.

Since the graph is a parabola opening upwards, we'll evaluate the given quadratic equation $y = x^2 - 6x + 8$ . Analyzing it and comparing leads to:

The vertex form can be rewritten or identified mathematically or visually from the given expression.
This quadratic formula aligns with prominent features of the parabola: its vertex, orientation, and intercepts, matching the graph.

Thus, as the parabola aligns perfectly with quadratic properties such as opening upwards, the formula that describes graph 1 correctly is:
$y = x^2 - 6x + 8$ .

Final Answer

$y=x^2-6x+8$

Key Points to Remember

Essential concepts to master this topic

Shape Recognition: Parabolas are U-shaped curves from quadratic functions
Technique: Check vertex and intercepts: $y = x^2 - 6x + 8$ opens upward
Verification: Test key points like vertex (3, -1) in equation ✓

Common Mistakes

Avoid these frequent errors

Confusing linear and quadratic graphs
Don't assume straight lines are always correct = wrong function type! Linear equations like y = 4x - 17 create straight lines, not curved parabolas. Always identify the curve shape first - parabolas mean quadratic functions.

Practice Quiz

Test your knowledge with interactive questions

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is
$$ f(x)<0 $$ true?

FAQ

Everything you need to know about this question

How can I tell if a graph shows a quadratic function?

Look for the U-shape or upside-down U! Quadratic functions always create parabolas - curved lines that bend. Linear functions make straight lines, while quadratic functions make curves.

What does the vertex of this parabola tell me?

The vertex is the lowest point (or highest if opening down). For $y = x^2 - 6x + 8$ , the vertex is at (3, -1), which matches the bottom of the curve in the graph.

Why are the other answer choices wrong?

Linear equations like y = 4x - 17 create straight lines, not curves. Only quadratic equations with $x^2$ terms can create the parabola shape shown in the graph.

How do I find where the parabola crosses the x-axis?

Set y = 0 and solve: $0 = x^2 - 6x + 8$ . Factor to get $(x-2)(x-4) = 0$ , so it crosses at x = 2 and x = 4.

What if I can't see the exact coordinates clearly?

Focus on the overall shape first! If it's curved like a U, it must be quadratic. Then look for general features like whether it opens up or down, and approximate where key points might be.

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