Determine Point C on the Quadratic Curve of f(x) = x² - 6x + 8

Q: What's the difference between x-intercept and y-intercept?

The y-intercept is where the graph crosses the y-axis (x = 0), while x-intercepts are where it crosses the x-axis (y = 0). For y-intercept, substitute x = 0. For x-intercepts, set f(x) = 0.

Q: Why does point C appear to be at the top of the parabola in the graph?

Point C is not at the top! The graph shows C on the y-axis at (0,8). The top of the parabola is the vertex, which occurs at x = 3 for this function.

Q: Can a quadratic function have more than one y-intercept?

No! Every function can have at most one y-intercept because there's only one point where x = 0. However, quadratics can have 0, 1, or 2 x-intercepts.

Y-Intercept with Quadratic Functions

The following function has been graphed below:

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

Calculate point C.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the coordinates of point C

00:03 Point C is the intersection point with the Y-axis of the function

00:07 At the intersection point with the Y-axis, X = 0

00:14 Substitute X=0 in the function and solve for Y

00:28 This is the Y value at point C

00:32 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

The following function has been graphed below:

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

Calculate point C.

Step-by-step solution

To solve for point C on the graph of the function $f(x) = x^2 - 6x + 8$ , we will determine the y-intercept of the graph.

According to the properties of a quadratic function, the y-intercept is found by evaluating the function at $x = 0$ . This provides the point where the graph crosses the y-axis.

Substituting $x = 0$ in the equation:

$f(0) = (0)^2 - 6 \times 0 + 8 = 8$

Thus, the y-coordinate of the intercept, and point C, is 8. Hence, point C is located at $(0, 8)$ .

Therefore, the solution to the problem is $(0, 8)$ .

Final Answer

$(0,8)$

Key Points to Remember

Essential concepts to master this topic

Y-Intercept Rule: Set x = 0 and evaluate f(0) to find where graph crosses y-axis
Technique: Substitute $x = 0$ into $f(x) = x^2 - 6x + 8$ gives f(0) = 8
Check: Point (0,8) should lie on the parabola when x = 0 ✓

Common Mistakes

Avoid these frequent errors

Finding x-intercepts instead of y-intercept
Don't set f(x) = 0 to solve for x-values = you get x-intercepts (2,0) and (4,0)! The y-intercept requires a different approach. Always substitute x = 0 into the function to find where it crosses the y-axis.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

$$ f(x)=-x^2+5x+6 $$

Calculate points A and B.

FAQ

Everything you need to know about this question

What's the difference between x-intercept and y-intercept?

The y-intercept is where the graph crosses the y-axis (x = 0), while x-intercepts are where it crosses the x-axis (y = 0). For y-intercept, substitute x = 0. For x-intercepts, set f(x) = 0.

Why does point C appear to be at the top of the parabola in the graph?

Point C is not at the top! The graph shows C on the y-axis at (0,8). The top of the parabola is the vertex, which occurs at x = 3 for this function.

How do I know which point on the graph is the y-intercept?

The y-intercept is always the point where the curve crosses the vertical y-axis. Look for where x = 0 on the graph - that's your y-intercept!

Can a quadratic function have more than one y-intercept?

No! Every function can have at most one y-intercept because there's only one point where x = 0. However, quadratics can have 0, 1, or 2 x-intercepts.

What if I get a negative y-intercept?

That's perfectly normal! The y-intercept can be positive, negative, or zero depending on the constant term in your quadratic function. Just substitute x = 0 and calculate.

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