Question

Graph the equation. $$ y=-x^{2}+8 x-15 $$

   Graph the equation.
$$
y=-x^{2}+8 x-15
$$

Intermediate Algebra

Intermediate Algebra

Lynn Marecek 2nd Edition

Chapter 11, Problem 130

Instant Answer

Solved by Expert Kathleen Carty

09/29/2022

Step 1

The vertex of the parabola is given by the formula $(-\frac{b}{2a}, f(-\frac{b}{2a}))$. In this case, $a = -1$ and $b = 8$. So, the x-coordinate of the vertex is $-\frac{8}{2(-1)} = 4$. Show more…

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Graph the equation. $$ y=-x^{2}+8 x-15 $$

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Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial typically written in the form ax² + bx + c. It is fundamental in algebra and represents the simplest form of a nonlinear function, exhibiting a parabolic graph. Understanding its structure, including the coefficients, is crucial for analyzing the behavior of the graph.

Parabola

The graph of a quadratic function is a parabola, a symmetrical curve that opens either upward or downward depending on the sign of the coefficient a. When a is negative, the parabola opens downward, indicating that the function attains a maximum value at its vertex.

Vertex

The vertex of a parabola is the highest or lowest point on its graph, acting as a turning point. It can be calculated using the formula x = -b/(2a) for the x-coordinate, with the corresponding y-coordinate found by substituting that value into the function. This point is critical for graphing and understanding the function's maximum or minimum.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. It plays an essential role in graphing quadratic functions by providing a reference line that reflects the inherent symmetry of the graph.

Intercepts

Intercepts are the points where the graph of the function crosses the coordinate axes. The y-intercept is found by evaluating the function at x = 0, while the x-intercepts (if they exist) are the solutions to the equation ax² + bx + c = 0. These intercepts provide key reference points that help in sketching the graph accurately.

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