Determine Coordinates: Calculating Point C on f(x) = x² - 6x + 5

Q: How do I know point C is the y-intercept?

Look at the graph! Point C is positioned on the y-axis (vertical line), which means its x-coordinate is 0. Any point on the y-axis has the form $ (0, y) $.

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

The y-intercept is where the graph crosses the y-axis (set x = 0). The x-intercepts are where the graph crosses the x-axis (set y = 0). This parabola has one y-intercept but two x-intercepts!

Q: Can I just read the y-intercept from the equation?

Yes! For any function $ f(x) = ax^2 + bx + c $, the y-intercept is simply c. In $ f(x) = x^2 - 6x + 5 $, c = 5, so the y-intercept is (0,5).

Q: What if I accidentally calculated the vertex instead?

The vertex formula gives $ x = -\frac{b}{2a} = 3 $, leading to point (3,-4). But point C is clearly on the y-axis, not at x = 3. Always check the graph location first!

Y-Intercept Identification with Quadratic Functions

The following function has been graphed below:

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

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 Y-axis, X = 0

00:11 Substitute X=0 in the function and solve to find the Y value

00:30 This is the Y value at point C

00:38 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:

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

Calculate point C.

Step-by-step solution

We are required to find a particular point C on the graph of the function $f(x) = x^2 - 6x + 5$ . Typically, important features of the graph include the vertex or y-intercept.

The function $f(x) = x^2 - 6x + 5$ is a quadratic function in standard form, where $a = 1$ , $b = -6$ , and $c = 5$ . Since point C is labeled near the y-axis, it is likely related to the y-intercept. The y-intercept is found by evaluating the function at $x = 0$ .

Calculate the y-intercept by substituting $x = 0$ into the function:

Substitute: $f(0) = (0)^2 - 6(0) + 5 = 5$

This gives the point on the y-axis where the function intersects, which is point C. Thus, point C is at the coordinates $(0, 5)$ .

This point matches choice 2 among the provided choices.

Therefore, the solution to the problem is $(0, 5)$ .

Final Answer

$(0,5)$

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
Substitution Method: f(0) = (0)² - 6(0) + 5 = 0 - 0 + 5 = 5
Verification: Point (0,5) should lie on y-axis and satisfy the original equation ✓

Common Mistakes

Avoid these frequent errors

Trying to find x-intercepts instead of y-intercept
Don't set f(x) = 0 when looking for point C on the y-axis = gives (1,0) and (5,0) instead! This finds where the parabola crosses the x-axis, not the y-axis. Always substitute x = 0 to find the y-intercept.

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

How do I know point C is the y-intercept?

Look at the graph! Point C is positioned on the y-axis (vertical line), which means its x-coordinate is 0. Any point on the y-axis has the form $(0, y)$ .

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

The y-intercept is where the graph crosses the y-axis (set x = 0). The x-intercepts are where the graph crosses the x-axis (set y = 0). This parabola has one y-intercept but two x-intercepts!

Can I just read the y-intercept from the equation?

Yes! For any function $f(x) = ax^2 + bx + c$ , the y-intercept is simply c. In $f(x) = x^2 - 6x + 5$ , c = 5, so the y-intercept is (0,5).

Why does substituting x = 0 work?

When x = 0, you're finding the function value at the y-axis. Since $0^2 = 0$ and $6 \times 0 = 0$ , only the constant term remains: f(0) = 5.

What if I accidentally calculated the vertex instead?

The vertex formula gives $x = -\frac{b}{2a} = 3$ , leading to point (3,-4). But point C is clearly on the y-axis, not at x = 3. Always check the graph location first!

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