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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 13th EditionCh. 3 - Polynomial and Rational Functions문제 17
4장, 문제 17

Match each function with its graph without actually entering it into a calculator. Then, after completing the exercises, check the answers with a calculator. Use the standard viewing window. ƒ(x) = (x + 4)2 - 3

검증된 단계별 안내
1
Identify the base function: recognize that the given function is a quadratic function in vertex form, which is \(f(x) = (x + 4)^2 - 3\). The base function is \(x^2\), a parabola opening upwards with vertex at the origin.
Determine the vertex of the parabola: since the function is in the form \(f(x) = (x - h)^2 + k\), the vertex is at \((-4, -3)\) because the expression inside the square is \((x + 4)\), which can be rewritten as \((x - (-4))\).
Analyze the transformations: the graph is shifted horizontally 4 units to the left (due to \(x + 4\)) and vertically 3 units down (due to \(-3\)). The parabola still opens upwards because the coefficient of the squared term is positive.
Sketch or visualize the graph based on these transformations: start with the basic parabola \(y = x^2\), move it left 4 units and down 3 units to place the vertex at \((-4, -3)\), and keep the shape the same (opening upwards).
Match the function to the graph that has a vertex at \((-4, -3)\) and opens upwards, without any stretching or reflecting, then verify your choice using a calculator with the standard viewing window.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Transformations of Quadratic Functions

Understanding how changes inside the function's formula affect its graph is essential. For f(x) = (x + 4)^2 - 3, the '+4' inside the parentheses shifts the graph horizontally left by 4 units, and the '-3' shifts it vertically down by 3 units. Recognizing these shifts helps match the function to its graph without graphing technology.
추천 영상:
4:22
Domain & Range of Transformed Functions

Standard Form and Vertex of a Parabola

A quadratic function in vertex form, f(x) = a(x - h)^2 + k, reveals the vertex directly as (h, k). Here, rewriting f(x) = (x + 4)^2 - 3 as (x - (-4))^2 - 3 shows the vertex at (-4, -3). Knowing the vertex location is key to identifying the parabola's position on the coordinate plane.
추천 영상:
04:34
Converting Standard Form to Vertex Form

Shape and Direction of Parabolas

The coefficient of the squared term determines the parabola's opening direction and width. Since the coefficient of (x + 4)^2 is positive 1, the parabola opens upward and has a standard width. This knowledge helps distinguish the graph from others that might open downward or be wider/narrower.
추천 영상:
5:28
Horizontal Parabolas
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